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Theorem infpssrlem3 7947
Description: Lemma for infpssr 7950. (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 6463 . . . 4  |-  ( rec ( `' F ,  C )  |`  om )  Fn  om
2 infpssrlem.e . . . . 5  |-  G  =  ( rec ( `' F ,  C )  |`  om )
32fneq1i 5354 . . . 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 5541 . . . . . 6  |-  ( c  =  (/)  ->  ( G `
 c )  =  ( G `  (/) ) )
76eleq1d 2362 . . . . 5  |-  ( c  =  (/)  ->  ( ( G `  c )  e.  A  <->  ( G `  (/) )  e.  A
) )
8 fveq2 5541 . . . . . 6  |-  ( c  =  b  ->  ( G `  c )  =  ( G `  b ) )
98eleq1d 2362 . . . . 5  |-  ( c  =  b  ->  (
( G `  c
)  e.  A  <->  ( G `  b )  e.  A
) )
10 fveq2 5541 . . . . . 6  |-  ( c  =  suc  b  -> 
( G `  c
)  =  ( G `
 suc  b )
)
1110eleq1d 2362 . . . . 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 7945 . . . . . 6  |-  ( ph  ->  ( G `  (/) )  =  C )
16 eldifi 3311 . . . . . . 7  |-  ( C  e.  ( A  \  B )  ->  C  e.  A )
1714, 16syl 15 . . . . . 6  |-  ( ph  ->  C  e.  A )
1815, 17eqeltrd 2370 . . . . 5  |-  ( ph  ->  ( G `  (/) )  e.  A )
1912adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( G `  b )  e.  A
)  ->  B  C_  A
)
20 f1ocnv 5501 . . . . . . . . . 10  |-  ( F : B -1-1-onto-> A  ->  `' F : A -1-1-onto-> B )
21 f1of 5488 . . . . . . . . . 10  |-  ( `' F : A -1-1-onto-> B  ->  `' F : A --> B )
2213, 20, 213syl 18 . . . . . . . . 9  |-  ( ph  ->  `' F : A --> B )
23 ffvelrn 5679 . . . . . . . . 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 3194 . . . . . . 7  |-  ( (
ph  /\  ( G `  b )  e.  A
)  ->  ( `' F `  ( G `  b ) )  e.  A )
2612, 13, 14, 2infpssrlem2 7946 . . . . . . . 8  |-  ( b  e.  om  ->  ( G `  suc  b )  =  ( `' F `  ( G `  b
) ) )
2726eleq1d 2362 . . . . . . 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 4700 . . . 4  |-  ( c  e.  om  ->  ( ph  ->  ( G `  c )  e.  A
) )
3130com12 27 . . 3  |-  ( ph  ->  ( c  e.  om  ->  ( G `  c
)  e.  A ) )
3231ralrimiv 2638 . 2  |-  ( ph  ->  A. c  e.  om  ( G `  c )  e.  A )
33 ffnfv 5701 . 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 1632    e. wcel 1696   A.wral 2556    \ cdif 3162    C_ wss 3165   (/)c0 3468   suc csuc 4410   omcom 4672   `'ccnv 4704    |` cres 4707    Fn wfn 5266   -->wf 5267   -1-1-onto->wf1o 5270   ` cfv 5271   reccrdg 6438
This theorem is referenced by:  infpssrlem4  7948  infpssrlem5  7949
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-recs 6404  df-rdg 6439
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