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Theorem infpssrlem3 7931
Description: Lemma for infpssr 7934. (Contributed by Stefan O'Rear, 30-Oct-2014.)
Hypotheses
Ref Expression
infpssrlem.a  |-  ( ph  ->  B  C_  A )
infpssrlem.c  |-  ( ph  ->  F : B -1-1-onto-> A )
infpssrlem.d  |-  ( ph  ->  C  e.  ( A 
\  B ) )
infpssrlem.e  |-  G  =  ( rec ( `' F ,  C )  |`  om )
Assertion
Ref Expression
infpssrlem3  |-  ( ph  ->  G : om --> A )

Proof of Theorem infpssrlem3
Dummy variables  b 
c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frfnom 6447 . . . 4  |-  ( rec ( `' F ,  C )  |`  om )  Fn  om
2 infpssrlem.e . . . . 5  |-  G  =  ( rec ( `' F ,  C )  |`  om )
32fneq1i 5338 . . . 4  |-  ( G  Fn  om  <->  ( rec ( `' F ,  C )  |`  om )  Fn  om )
41, 3mpbir 200 . . 3  |-  G  Fn  om
54a1i 10 . 2  |-  ( ph  ->  G  Fn  om )
6 fveq2 5525 . . . . . 6  |-  ( c  =  (/)  ->  ( G `
 c )  =  ( G `  (/) ) )
76eleq1d 2349 . . . . 5  |-  ( c  =  (/)  ->  ( ( G `  c )  e.  A  <->  ( G `  (/) )  e.  A
) )
8 fveq2 5525 . . . . . 6  |-  ( c  =  b  ->  ( G `  c )  =  ( G `  b ) )
98eleq1d 2349 . . . . 5  |-  ( c  =  b  ->  (
( G `  c
)  e.  A  <->  ( G `  b )  e.  A
) )
10 fveq2 5525 . . . . . 6  |-  ( c  =  suc  b  -> 
( G `  c
)  =  ( G `
 suc  b )
)
1110eleq1d 2349 . . . . 5  |-  ( c  =  suc  b  -> 
( ( G `  c )  e.  A  <->  ( G `  suc  b
)  e.  A ) )
12 infpssrlem.a . . . . . . 7  |-  ( ph  ->  B  C_  A )
13 infpssrlem.c . . . . . . 7  |-  ( ph  ->  F : B -1-1-onto-> A )
14 infpssrlem.d . . . . . . 7  |-  ( ph  ->  C  e.  ( A 
\  B ) )
1512, 13, 14, 2infpssrlem1 7929 . . . . . 6  |-  ( ph  ->  ( G `  (/) )  =  C )
16 eldifi 3298 . . . . . . 7  |-  ( C  e.  ( A  \  B )  ->  C  e.  A )
1714, 16syl 15 . . . . . 6  |-  ( ph  ->  C  e.  A )
1815, 17eqeltrd 2357 . . . . 5  |-  ( ph  ->  ( G `  (/) )  e.  A )
1912adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( G `  b )  e.  A
)  ->  B  C_  A
)
20 f1ocnv 5485 . . . . . . . . . 10  |-  ( F : B -1-1-onto-> A  ->  `' F : A -1-1-onto-> B )
21 f1of 5472 . . . . . . . . . 10  |-  ( `' F : A -1-1-onto-> B  ->  `' F : A --> B )
2213, 20, 213syl 18 . . . . . . . . 9  |-  ( ph  ->  `' F : A --> B )
23 ffvelrn 5663 . . . . . . . . 9  |-  ( ( `' F : A --> B  /\  ( G `  b )  e.  A )  -> 
( `' F `  ( G `  b ) )  e.  B )
2422, 23sylan 457 . . . . . . . 8  |-  ( (
ph  /\  ( G `  b )  e.  A
)  ->  ( `' F `  ( G `  b ) )  e.  B )
2519, 24sseldd 3181 . . . . . . 7  |-  ( (
ph  /\  ( G `  b )  e.  A
)  ->  ( `' F `  ( G `  b ) )  e.  A )
2612, 13, 14, 2infpssrlem2 7930 . . . . . . . 8  |-  ( b  e.  om  ->  ( G `  suc  b )  =  ( `' F `  ( G `  b
) ) )
2726eleq1d 2349 . . . . . . 7  |-  ( b  e.  om  ->  (
( G `  suc  b )  e.  A  <->  ( `' F `  ( G `
 b ) )  e.  A ) )
2825, 27syl5ibr 212 . . . . . 6  |-  ( b  e.  om  ->  (
( ph  /\  ( G `  b )  e.  A )  ->  ( G `  suc  b )  e.  A ) )
2928exp3a 425 . . . . 5  |-  ( b  e.  om  ->  ( ph  ->  ( ( G `
 b )  e.  A  ->  ( G `  suc  b )  e.  A ) ) )
307, 9, 11, 18, 29finds2 4684 . . . 4  |-  ( c  e.  om  ->  ( ph  ->  ( G `  c )  e.  A
) )
3130com12 27 . . 3  |-  ( ph  ->  ( c  e.  om  ->  ( G `  c
)  e.  A ) )
3231ralrimiv 2625 . 2  |-  ( ph  ->  A. c  e.  om  ( G `  c )  e.  A )
33 ffnfv 5685 . 2  |-  ( G : om --> A  <->  ( G  Fn  om  /\  A. c  e.  om  ( G `  c )  e.  A
) )
345, 32, 33sylanbrc 645 1  |-  ( ph  ->  G : om --> A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    \ cdif 3149    C_ wss 3152   (/)c0 3455   suc csuc 4394   omcom 4656   `'ccnv 4688    |` cres 4691    Fn wfn 5250   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255   reccrdg 6422
This theorem is referenced by:  infpssrlem4  7932  infpssrlem5  7933
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-reu 2550  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-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  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-lim 4397  df-suc 4398  df-om 4657  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-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-recs 6388  df-rdg 6423
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