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Theorem tfrlem2 6392
Description: Lemma for transfinite recursion. This provides some messy details needed to link tfrlem1 6391 into the main proof. (Contributed by NM, 23-Mar-1995.) (Revised by David Abernethy, 19-Jun-2012.)
Assertion
Ref Expression
tfrlem2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  G )  -> 
( A  e.  On  ->  ( A. w ( A  e.  On  ->  ( w  e.  A  -> 
( ( F `  w )  =  ( B `  ( F  |`  w ) )  /\  ( G `  w )  =  ( B `  ( G  |`  w ) ) ) ) )  ->  y  =  z ) ) ) )
Distinct variable groups:    w, A    w, F    w, G    x, w
Allowed substitution hints:    A( x, y, z)    B( x, y, z, w)    F( x, y, z)    G( x, y, z)

Proof of Theorem tfrlem2
StepHypRef Expression
1 abai 770 . . . . 5  |-  ( ( A  e.  On  /\  ( w  e.  A  ->  ( ( F `  w )  =  ( B `  ( F  |`  w ) )  /\  ( G `  w )  =  ( B `  ( G  |`  w ) ) ) ) )  <-> 
( A  e.  On  /\  ( A  e.  On  ->  ( w  e.  A  ->  ( ( F `  w )  =  ( B `  ( F  |`  w ) )  /\  ( G `  w )  =  ( B `  ( G  |`  w ) ) ) ) ) ) )
21albii 1553 . . . 4  |-  ( A. w ( A  e.  On  /\  ( w  e.  A  ->  (
( F `  w
)  =  ( B `
 ( F  |`  w ) )  /\  ( G `  w )  =  ( B `  ( G  |`  w ) ) ) ) )  <->  A. w ( A  e.  On  /\  ( A  e.  On  ->  (
w  e.  A  -> 
( ( F `  w )  =  ( B `  ( F  |`  w ) )  /\  ( G `  w )  =  ( B `  ( G  |`  w ) ) ) ) ) ) )
3 19.28v 1836 . . . 4  |-  ( A. w ( A  e.  On  /\  ( w  e.  A  ->  (
( F `  w
)  =  ( B `
 ( F  |`  w ) )  /\  ( G `  w )  =  ( B `  ( G  |`  w ) ) ) ) )  <-> 
( A  e.  On  /\ 
A. w ( w  e.  A  ->  (
( F `  w
)  =  ( B `
 ( F  |`  w ) )  /\  ( G `  w )  =  ( B `  ( G  |`  w ) ) ) ) ) )
4 19.28v 1836 . . . 4  |-  ( A. w ( A  e.  On  /\  ( A  e.  On  ->  (
w  e.  A  -> 
( ( F `  w )  =  ( B `  ( F  |`  w ) )  /\  ( G `  w )  =  ( B `  ( G  |`  w ) ) ) ) ) )  <->  ( A  e.  On  /\  A. w
( A  e.  On  ->  ( w  e.  A  ->  ( ( F `  w )  =  ( B `  ( F  |`  w ) )  /\  ( G `  w )  =  ( B `  ( G  |`  w ) ) ) ) ) ) )
52, 3, 43bitr3ri 267 . . 3  |-  ( ( A  e.  On  /\  A. w ( A  e.  On  ->  ( w  e.  A  ->  ( ( F `  w )  =  ( B `  ( F  |`  w ) )  /\  ( G `
 w )  =  ( B `  ( G  |`  w ) ) ) ) ) )  <-> 
( A  e.  On  /\ 
A. w ( w  e.  A  ->  (
( F `  w
)  =  ( B `
 ( F  |`  w ) )  /\  ( G `  w )  =  ( B `  ( G  |`  w ) ) ) ) ) )
6 df-ral 2548 . . . . . 6  |-  ( A. w  e.  A  (
( F `  w
)  =  ( B `
 ( F  |`  w ) )  /\  ( G `  w )  =  ( B `  ( G  |`  w ) ) )  <->  A. w
( w  e.  A  ->  ( ( F `  w )  =  ( B `  ( F  |`  w ) )  /\  ( G `  w )  =  ( B `  ( G  |`  w ) ) ) ) )
76anbi2i 675 . . . . 5  |-  ( ( A  e.  On  /\  A. w  e.  A  ( ( F `  w
)  =  ( B `
 ( F  |`  w ) )  /\  ( G `  w )  =  ( B `  ( G  |`  w ) ) ) )  <->  ( A  e.  On  /\  A. w
( w  e.  A  ->  ( ( F `  w )  =  ( B `  ( F  |`  w ) )  /\  ( G `  w )  =  ( B `  ( G  |`  w ) ) ) ) ) )
8 fnop 5347 . . . . . . . . 9  |-  ( ( F  Fn  A  /\  <.
x ,  y >.  e.  F )  ->  x  e.  A )
98adantlr 695 . . . . . . . 8  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  <. x ,  y >.  e.  F
)  ->  x  e.  A )
10 tfrlem1 6391 . . . . . . . . . . 11  |-  ( A  e.  On  ->  (
( F  Fn  A  /\  G  Fn  A
)  ->  ( A. w  e.  A  (
( F `  w
)  =  ( B `
 ( F  |`  w ) )  /\  ( G `  w )  =  ( B `  ( G  |`  w ) ) )  ->  A. w  e.  A  ( F `  w )  =  ( G `  w ) ) ) )
1110com12 27 . . . . . . . . . 10  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( A  e.  On  ->  ( A. w  e.  A  ( ( F `
 w )  =  ( B `  ( F  |`  w ) )  /\  ( G `  w )  =  ( B `  ( G  |`  w ) ) )  ->  A. w  e.  A  ( F `  w )  =  ( G `  w ) ) ) )
1211imp3a 420 . . . . . . . . 9  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( ( A  e.  On  /\  A. w  e.  A  ( ( F `  w )  =  ( B `  ( F  |`  w ) )  /\  ( G `
 w )  =  ( B `  ( G  |`  w ) ) ) )  ->  A. w  e.  A  ( F `  w )  =  ( G `  w ) ) )
1312adantr 451 . . . . . . . 8  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  <. x ,  y >.  e.  F
)  ->  ( ( A  e.  On  /\  A. w  e.  A  (
( F `  w
)  =  ( B `
 ( F  |`  w ) )  /\  ( G `  w )  =  ( B `  ( G  |`  w ) ) ) )  ->  A. w  e.  A  ( F `  w )  =  ( G `  w ) ) )
14 fveq2 5525 . . . . . . . . . 10  |-  ( w  =  x  ->  ( F `  w )  =  ( F `  x ) )
15 fveq2 5525 . . . . . . . . . 10  |-  ( w  =  x  ->  ( G `  w )  =  ( G `  x ) )
1614, 15eqeq12d 2297 . . . . . . . . 9  |-  ( w  =  x  ->  (
( F `  w
)  =  ( G `
 w )  <->  ( F `  x )  =  ( G `  x ) ) )
1716rspcv 2880 . . . . . . . 8  |-  ( x  e.  A  ->  ( A. w  e.  A  ( F `  w )  =  ( G `  w )  ->  ( F `  x )  =  ( G `  x ) ) )
189, 13, 17sylsyld 52 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  <. x ,  y >.  e.  F
)  ->  ( ( A  e.  On  /\  A. w  e.  A  (
( F `  w
)  =  ( B `
 ( F  |`  w ) )  /\  ( G `  w )  =  ( B `  ( G  |`  w ) ) ) )  -> 
( F `  x
)  =  ( G `
 x ) ) )
1918imp 418 . . . . . 6  |-  ( ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  <. x ,  y >.  e.  F
)  /\  ( A  e.  On  /\  A. w  e.  A  ( ( F `  w )  =  ( B `  ( F  |`  w ) )  /\  ( G `
 w )  =  ( B `  ( G  |`  w ) ) ) ) )  -> 
( F `  x
)  =  ( G `
 x ) )
2019adantlrr 701 . . . . 5  |-  ( ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  G ) )  /\  ( A  e.  On  /\  A. w  e.  A  ( ( F `  w )  =  ( B `  ( F  |`  w ) )  /\  ( G `
 w )  =  ( B `  ( G  |`  w ) ) ) ) )  -> 
( F `  x
)  =  ( G `
 x ) )
217, 20sylan2br 462 . . . 4  |-  ( ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  G ) )  /\  ( A  e.  On  /\  A. w
( w  e.  A  ->  ( ( F `  w )  =  ( B `  ( F  |`  w ) )  /\  ( G `  w )  =  ( B `  ( G  |`  w ) ) ) ) ) )  ->  ( F `  x )  =  ( G `  x ) )
22 fnfun 5341 . . . . . . . 8  |-  ( F  Fn  A  ->  Fun  F )
23 fnfun 5341 . . . . . . . 8  |-  ( G  Fn  A  ->  Fun  G )
2422, 23anim12i 549 . . . . . . 7  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( Fun  F  /\  Fun  G ) )
25 funopfv 5562 . . . . . . . . . 10  |-  ( Fun 
F  ->  ( <. x ,  y >.  e.  F  ->  ( F `  x
)  =  y ) )
2625imp 418 . . . . . . . . 9  |-  ( ( Fun  F  /\  <. x ,  y >.  e.  F
)  ->  ( F `  x )  =  y )
27 funopfv 5562 . . . . . . . . . 10  |-  ( Fun 
G  ->  ( <. x ,  z >.  e.  G  ->  ( G `  x
)  =  z ) )
2827imp 418 . . . . . . . . 9  |-  ( ( Fun  G  /\  <. x ,  z >.  e.  G
)  ->  ( G `  x )  =  z )
2926, 28anim12i 549 . . . . . . . 8  |-  ( ( ( Fun  F  /\  <.
x ,  y >.  e.  F )  /\  ( Fun  G  /\  <. x ,  z >.  e.  G
) )  ->  (
( F `  x
)  =  y  /\  ( G `  x )  =  z ) )
3029an4s 799 . . . . . . 7  |-  ( ( ( Fun  F  /\  Fun  G )  /\  ( <. x ,  y >.  e.  F  /\  <. x ,  z >.  e.  G
) )  ->  (
( F `  x
)  =  y  /\  ( G `  x )  =  z ) )
3124, 30sylan 457 . . . . . 6  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  G ) )  ->  ( ( F `
 x )  =  y  /\  ( G `
 x )  =  z ) )
32 eqeq12 2295 . . . . . 6  |-  ( ( ( F `  x
)  =  y  /\  ( G `  x )  =  z )  -> 
( ( F `  x )  =  ( G `  x )  <-> 
y  =  z ) )
3331, 32syl 15 . . . . 5  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  G ) )  ->  ( ( F `
 x )  =  ( G `  x
)  <->  y  =  z ) )
3433adantr 451 . . . 4  |-  ( ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  G ) )  /\  ( A  e.  On  /\  A. w
( w  e.  A  ->  ( ( F `  w )  =  ( B `  ( F  |`  w ) )  /\  ( G `  w )  =  ( B `  ( G  |`  w ) ) ) ) ) )  ->  ( ( F `  x )  =  ( G `  x )  <->  y  =  z ) )
3521, 34mpbid 201 . . 3  |-  ( ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  G ) )  /\  ( A  e.  On  /\  A. w
( w  e.  A  ->  ( ( F `  w )  =  ( B `  ( F  |`  w ) )  /\  ( G `  w )  =  ( B `  ( G  |`  w ) ) ) ) ) )  ->  y  =  z )
365, 35sylan2b 461 . 2  |-  ( ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  G ) )  /\  ( A  e.  On  /\  A. w
( A  e.  On  ->  ( w  e.  A  ->  ( ( F `  w )  =  ( B `  ( F  |`  w ) )  /\  ( G `  w )  =  ( B `  ( G  |`  w ) ) ) ) ) ) )  ->  y  =  z )
3736exp43 595 1  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  G )  -> 
( A  e.  On  ->  ( A. w ( A  e.  On  ->  ( w  e.  A  -> 
( ( F `  w )  =  ( B `  ( F  |`  w ) )  /\  ( G `  w )  =  ( B `  ( G  |`  w ) ) ) ) )  ->  y  =  z ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527    = wceq 1623    e. wcel 1684   A.wral 2543   <.cop 3643   Oncon0 4392    |` cres 4691   Fun wfun 5249    Fn wfn 5250   ` cfv 5255
This theorem is referenced by:  tfrlem5  6396
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263
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