Theorem infpssrlem3 9127

Description: Lemma for infpssr 9130. (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.

Step	Hyp	Ref	Expression
1		frfnom 7530	. . . 4 |- ( rec ( `' F , C ) |` om ) Fn om
2		infpssrlem.e	. . . . 5 |- G = ( rec ( `' F , C ) |` om )
3	2	fneq1i 5985	. . . 4 |- ( G Fn om <-> ( rec ( `' F , C ) |` om ) Fn om )
4	1, 3	mpbir 221	. . 3 |- G Fn om
5	4	a1i 11	. 2 |- ( ph -> G Fn om )
6		fveq2 6191	. . . . . 6 |- ( c = (/) -> ( G ` c ) = ( G ` (/) ) )
7	6	eleq1d 2686	. . . . 5 |- ( c = (/) -> ( ( G ` c ) e. A <-> ( G ` (/) ) e. A ) )
8		fveq2 6191	. . . . . 6 |- ( c = b -> ( G ` c ) = ( G ` b ) )
9	8	eleq1d 2686	. . . . 5 |- ( c = b -> ( ( G ` c ) e. A <-> ( G ` b ) e. A ) )
10		fveq2 6191	. . . . . 6 |- ( c = suc b -> ( G ` c ) = ( G ` suc b ) )
11	10	eleq1d 2686	. . . . 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 ) )
15	12, 13, 14, 2	infpssrlem1 9125	. . . . . 6 |- ( ph -> ( G ` (/) ) = C )
16	14	eldifad 3586	. . . . . 6 |- ( ph -> C e. A )
17	15, 16	eqeltrd 2701	. . . . 5 |- ( ph -> ( G ` (/) ) e. A )
18	12	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( G ` b ) e. A ) -> B C_ A )
19		f1ocnv 6149	. . . . . . . . . 10 |- ( F : B -1-1-onto-> A -> `' F : A -1-1-onto-> B )
20		f1of 6137	. . . . . . . . . 10 |- ( `' F : A -1-1-onto-> B -> `' F : A --> B )
21	13, 19, 20	3syl 18	. . . . . . . . 9 |- ( ph -> `' F : A --> B )
22	21	ffvelrnda 6359	. . . . . . . 8 |- ( ( ph /\ ( G ` b ) e. A ) -> ( `' F ` ( G ` b ) ) e. B )
23	18, 22	sseldd 3604	. . . . . . 7 |- ( ( ph /\ ( G ` b ) e. A ) -> ( `' F ` ( G ` b ) ) e. A )
24	12, 13, 14, 2	infpssrlem2 9126	. . . . . . . 8 |- ( b e. om -> ( G ` suc b ) = ( `' F ` ( G ` b ) ) )
25	24	eleq1d 2686	. . . . . . 7 |- ( b e. om -> ( ( G ` suc b ) e. A <-> ( `' F ` ( G ` b ) ) e. A ) )
26	23, 25	syl5ibr 236	. . . . . 6 |- ( b e. om -> ( ( ph /\ ( G ` b ) e. A ) -> ( G ` suc b ) e. A ) )
27	26	expd 452	. . . . 5 |- ( b e. om -> ( ph -> ( ( G ` b ) e. A -> ( G ` suc b ) e. A ) ) )
28	7, 9, 11, 17, 27	finds2 7094	. . . 4 |- ( c e. om -> ( ph -> ( G ` c ) e. A ) )
29	28	com12 32	. . 3 |- ( ph -> ( c e. om -> ( G ` c ) e. A ) )
30	29	ralrimiv 2965	. 2 |- ( ph -> A. c e. om ( G ` c ) e. A )
31		ffnfv 6388	. 2 |- ( G : om --> A <-> ( G Fn om /\ A. c e. om ( G ` c ) e. A ) )
32	5, 30, 31	sylanbrc 698	1 |- ( ph -> G : om --> A )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   \ cdif 3571   C_ wss 3574   (/)c0 3915   `'ccnv 5113   |` cres 5116   suc csuc 5725   Fn wfn 5883   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888   omcom 7065   reccrdg 7505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506

This theorem is referenced by:  infpssrlem4  9128  infpssrlem5  9129