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Theorem infpssrlem3 8185
Description: Lemma for infpssr 8188. (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 6692 . . . 4  |-  ( rec ( `' F ,  C )  |`  om )  Fn  om
2 infpssrlem.e . . . . 5  |-  G  =  ( rec ( `' F ,  C )  |`  om )
32fneq1i 5539 . . . 4  |-  ( G  Fn  om  <->  ( rec ( `' F ,  C )  |`  om )  Fn  om )
41, 3mpbir 201 . . 3  |-  G  Fn  om
54a1i 11 . 2  |-  ( ph  ->  G  Fn  om )
6 fveq2 5728 . . . . . 6  |-  ( c  =  (/)  ->  ( G `
 c )  =  ( G `  (/) ) )
76eleq1d 2502 . . . . 5  |-  ( c  =  (/)  ->  ( ( G `  c )  e.  A  <->  ( G `  (/) )  e.  A
) )
8 fveq2 5728 . . . . . 6  |-  ( c  =  b  ->  ( G `  c )  =  ( G `  b ) )
98eleq1d 2502 . . . . 5  |-  ( c  =  b  ->  (
( G `  c
)  e.  A  <->  ( G `  b )  e.  A
) )
10 fveq2 5728 . . . . . 6  |-  ( c  =  suc  b  -> 
( G `  c
)  =  ( G `
 suc  b )
)
1110eleq1d 2502 . . . . 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 8183 . . . . . 6  |-  ( ph  ->  ( G `  (/) )  =  C )
1614eldifad 3332 . . . . . 6  |-  ( ph  ->  C  e.  A )
1715, 16eqeltrd 2510 . . . . 5  |-  ( ph  ->  ( G `  (/) )  e.  A )
1812adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( G `  b )  e.  A
)  ->  B  C_  A
)
19 f1ocnv 5687 . . . . . . . . . 10  |-  ( F : B -1-1-onto-> A  ->  `' F : A -1-1-onto-> B )
20 f1of 5674 . . . . . . . . . 10  |-  ( `' F : A -1-1-onto-> B  ->  `' F : A --> B )
2113, 19, 203syl 19 . . . . . . . . 9  |-  ( ph  ->  `' F : A --> B )
2221ffvelrnda 5870 . . . . . . . 8  |-  ( (
ph  /\  ( G `  b )  e.  A
)  ->  ( `' F `  ( G `  b ) )  e.  B )
2318, 22sseldd 3349 . . . . . . 7  |-  ( (
ph  /\  ( G `  b )  e.  A
)  ->  ( `' F `  ( G `  b ) )  e.  A )
2412, 13, 14, 2infpssrlem2 8184 . . . . . . . 8  |-  ( b  e.  om  ->  ( G `  suc  b )  =  ( `' F `  ( G `  b
) ) )
2524eleq1d 2502 . . . . . . 7  |-  ( b  e.  om  ->  (
( G `  suc  b )  e.  A  <->  ( `' F `  ( G `
 b ) )  e.  A ) )
2623, 25syl5ibr 213 . . . . . 6  |-  ( b  e.  om  ->  (
( ph  /\  ( G `  b )  e.  A )  ->  ( G `  suc  b )  e.  A ) )
2726exp3a 426 . . . . 5  |-  ( b  e.  om  ->  ( ph  ->  ( ( G `
 b )  e.  A  ->  ( G `  suc  b )  e.  A ) ) )
287, 9, 11, 17, 27finds2 4873 . . . 4  |-  ( c  e.  om  ->  ( ph  ->  ( G `  c )  e.  A
) )
2928com12 29 . . 3  |-  ( ph  ->  ( c  e.  om  ->  ( G `  c
)  e.  A ) )
3029ralrimiv 2788 . 2  |-  ( ph  ->  A. c  e.  om  ( G `  c )  e.  A )
31 ffnfv 5894 . 2  |-  ( G : om --> A  <->  ( G  Fn  om  /\  A. c  e.  om  ( G `  c )  e.  A
) )
325, 30, 31sylanbrc 646 1  |-  ( ph  ->  G : om --> A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705    \ cdif 3317    C_ wss 3320   (/)c0 3628   suc csuc 4583   omcom 4845   `'ccnv 4877    |` cres 4880    Fn wfn 5449   -->wf 5450   -1-1-onto->wf1o 5453   ` cfv 5454   reccrdg 6667
This theorem is referenced by:  infpssrlem4  8186  infpssrlem5  8187
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-recs 6633  df-rdg 6668
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