Theorem smfpimcclem 41013

Description: Lemma for smfpimcc 41014 given the choice function C. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
smfpimcclem.n	|- F/ n ph
smfpimcclem.z	|- Z e. V
smfpimcclem.s	|- ( ph -> S e. W )
smfpimcclem.c	|- ( ( ph /\ y e. ran ( n e. Z |-> { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } ) ) -> ( C ` y ) e. y )
smfpimcclem.h	|- H = ( n e. Z |-> ( C ` { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } ) )

Assertion

Ref	Expression
smfpimcclem	|- ( ph -> E. h ( h : Z --> S /\ A. n e. Z ( `' ( F ` n ) " A ) = ( ( h ` n ) i^i dom ( F ` n ) ) ) )

Distinct variable groups:   A, h   A, s, y   C, s, y   h, F   F, s, y   h, H   S, h, n   S, s, y, n   h, Z, n   y, Z   ph, y

Allowed substitution hints:   ph( h, n, s)   A( n)   C( h, n)   F( n)   H( y, n, s)   V( y, h, n, s)   W( y, h, n, s)   Z( s)

Proof of Theorem smfpimcclem

Step	Hyp	Ref	Expression
1		smfpimcclem.n	. . 3 |- F/ n ph
2		nfcv 2764	. . . . 5 |- F/_ s S
3	2	ssrab2f 39300	. . . 4 |- { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } C_ S
4		eqid 2622	. . . . . . 7 |- { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } = { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) }
5		smfpimcclem.s	. . . . . . 7 |- ( ph -> S e. W )
6	4, 5	rabexd 4814	. . . . . 6 |- ( ph -> { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } e. _V )
7	6	adantr 481	. . . . 5 |- ( ( ph /\ n e. Z ) -> { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } e. _V )
8		simpl 473	. . . . . 6 |- ( ( ph /\ n e. Z ) -> ph )
9		simpr 477	. . . . . . 7 |- ( ( ph /\ n e. Z ) -> n e. Z )
10		eqid 2622	. . . . . . . 8 |- ( n e. Z |-> { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } ) = ( n e. Z |-> { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } )
11	10	elrnmpt1 5374	. . . . . . 7 |- ( ( n e. Z /\ { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } e. _V ) -> { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } e. ran ( n e. Z |-> { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } ) )
12	9, 7, 11	syl2anc 693	. . . . . 6 |- ( ( ph /\ n e. Z ) -> { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } e. ran ( n e. Z |-> { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } ) )
13	8, 12	jca 554	. . . . 5 |- ( ( ph /\ n e. Z ) -> ( ph /\ { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } e. ran ( n e. Z |-> { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } ) ) )
14		eleq1 2689	. . . . . . . 8 |- ( y = { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } -> ( y e. ran ( n e. Z |-> { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } ) <-> { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } e. ran ( n e. Z |-> { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } ) ) )
15	14	anbi2d 740	. . . . . . 7 |- ( y = { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } -> ( ( ph /\ y e. ran ( n e. Z |-> { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } ) ) <-> ( ph /\ { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } e. ran ( n e. Z |-> { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } ) ) ) )
16		fveq2 6191	. . . . . . . 8 |- ( y = { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } -> ( C ` y ) = ( C ` { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } ) )
17		id 22	. . . . . . . 8 |- ( y = { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } -> y = { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } )
18	16, 17	eleq12d 2695	. . . . . . 7 |- ( y = { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } -> ( ( C ` y ) e. y <-> ( C ` { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } ) e. { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } ) )
19	15, 18	imbi12d 334	. . . . . 6 |- ( y = { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } -> ( ( ( ph /\ y e. ran ( n e. Z |-> { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } ) ) -> ( C ` y ) e. y ) <-> ( ( ph /\ { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } e. ran ( n e. Z |-> { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } ) ) -> ( C ` { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } ) e. { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } ) ) )
20		smfpimcclem.c	. . . . . 6 |- ( ( ph /\ y e. ran ( n e. Z |-> { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } ) ) -> ( C ` y ) e. y )
21	19, 20	vtoclg 3266	. . . . 5 |- ( { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } e. _V -> ( ( ph /\ { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } e. ran ( n e. Z |-> { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } ) ) -> ( C ` { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } ) e. { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } ) )
22	7, 13, 21	sylc 65	. . . 4 |- ( ( ph /\ n e. Z ) -> ( C ` { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } ) e. { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } )
23	3, 22	sseldi 3601	. . 3 |- ( ( ph /\ n e. Z ) -> ( C ` { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } ) e. S )
24		smfpimcclem.h	. . 3 |- H = ( n e. Z |-> ( C ` { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } ) )
25	1, 23, 24	fmptdf 6387	. 2 |- ( ph -> H : Z --> S )
26		nfcv 2764	. . . . . . . . 9 |- F/_ s C
27		nfrab1 3122	. . . . . . . . 9 |- F/_ s { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) }
28	26, 27	nffv 6198	. . . . . . . 8 |- F/_ s ( C ` { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } )
29		nfcv 2764	. . . . . . . . 9 |- F/_ s ( `' ( F ` n ) " A )
30		nfcv 2764	. . . . . . . . . 10 |- F/_ s dom ( F ` n )
31	28, 30	nfin 3820	. . . . . . . . 9 |- F/_ s ( ( C ` { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } ) i^i dom ( F ` n ) )
32	29, 31	nfeq 2776	. . . . . . . 8 |- F/ s ( `' ( F ` n ) " A ) = ( ( C ` { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } ) i^i dom ( F ` n ) )
33		ineq1 3807	. . . . . . . . 9 |- ( s = ( C ` { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } ) -> ( s i^i dom ( F ` n ) ) = ( ( C ` { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } ) i^i dom ( F ` n ) ) )
34	33	eqeq2d 2632	. . . . . . . 8 |- ( s = ( C ` { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } ) -> ( ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) <-> ( `' ( F ` n ) " A ) = ( ( C ` { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } ) i^i dom ( F ` n ) ) ) )
35	28, 2, 32, 34	elrabf 3360	. . . . . . 7 |- ( ( C ` { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } ) e. { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } <-> ( ( C ` { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } ) e. S /\ ( `' ( F ` n ) " A ) = ( ( C ` { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } ) i^i dom ( F ` n ) ) ) )
36	22, 35	sylib 208	. . . . . 6 |- ( ( ph /\ n e. Z ) -> ( ( C ` { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } ) e. S /\ ( `' ( F ` n ) " A ) = ( ( C ` { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } ) i^i dom ( F ` n ) ) ) )
37	36	simprd 479	. . . . 5 |- ( ( ph /\ n e. Z ) -> ( `' ( F ` n ) " A ) = ( ( C ` { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } ) i^i dom ( F ` n ) ) )
38	24	a1i 11	. . . . . . 7 |- ( ph -> H = ( n e. Z |-> ( C ` { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } ) ) )
39	22	elexd 3214	. . . . . . 7 |- ( ( ph /\ n e. Z ) -> ( C ` { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } ) e. _V )
40	38, 39	fvmpt2d 6293	. . . . . 6 |- ( ( ph /\ n e. Z ) -> ( H ` n ) = ( C ` { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } ) )
41	40	ineq1d 3813	. . . . 5 |- ( ( ph /\ n e. Z ) -> ( ( H ` n ) i^i dom ( F ` n ) ) = ( ( C ` { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } ) i^i dom ( F ` n ) ) )
42	37, 41	eqtr4d 2659	. . . 4 |- ( ( ph /\ n e. Z ) -> ( `' ( F ` n ) " A ) = ( ( H ` n ) i^i dom ( F ` n ) ) )
43	42	ex 450	. . 3 |- ( ph -> ( n e. Z -> ( `' ( F ` n ) " A ) = ( ( H ` n ) i^i dom ( F ` n ) ) ) )
44	1, 43	ralrimi 2957	. 2 |- ( ph -> A. n e. Z ( `' ( F ` n ) " A ) = ( ( H ` n ) i^i dom ( F ` n ) ) )
45		smfpimcclem.z	. . . . . 6 |- Z e. V
46	45	elexi 3213	. . . . 5 |- Z e. _V
47	46	mptex 6486	. . . 4 |- ( n e. Z |-> ( C ` { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } ) ) e. _V
48	24, 47	eqeltri 2697	. . 3 |- H e. _V
49		feq1 6026	. . . 4 |- ( h = H -> ( h : Z --> S <-> H : Z --> S ) )
50		nfcv 2764	. . . . . 6 |- F/_ n h
51		nfmpt1 4747	. . . . . . 7 |- F/_ n ( n e. Z |-> ( C ` { s e. S | ( `' ( F ` n ) " A ) = ( s i^i dom ( F ` n ) ) } ) )
52	24, 51	nfcxfr 2762	. . . . . 6 |- F/_ n H
53	50, 52	nfeq 2776	. . . . 5 |- F/ n h = H
54		fveq1 6190	. . . . . . 7 |- ( h = H -> ( h ` n ) = ( H ` n ) )
55	54	ineq1d 3813	. . . . . 6 |- ( h = H -> ( ( h ` n ) i^i dom ( F ` n ) ) = ( ( H ` n ) i^i dom ( F ` n ) ) )
56	55	eqeq2d 2632	. . . . 5 |- ( h = H -> ( ( `' ( F ` n ) " A ) = ( ( h ` n ) i^i dom ( F ` n ) ) <-> ( `' ( F ` n ) " A ) = ( ( H ` n ) i^i dom ( F ` n ) ) ) )
57	53, 56	ralbid 2983	. . . 4 |- ( h = H -> ( A. n e. Z ( `' ( F ` n ) " A ) = ( ( h ` n ) i^i dom ( F ` n ) ) <-> A. n e. Z ( `' ( F ` n ) " A ) = ( ( H ` n ) i^i dom ( F ` n ) ) ) )
58	49, 57	anbi12d 747	. . 3 |- ( h = H -> ( ( h : Z --> S /\ A. n e. Z ( `' ( F ` n ) " A ) = ( ( h ` n ) i^i dom ( F ` n ) ) ) <-> ( H : Z --> S /\ A. n e. Z ( `' ( F ` n ) " A ) = ( ( H ` n ) i^i dom ( F ` n ) ) ) ) )
59	48, 58	spcev 3300	. 2 |- ( ( H : Z --> S /\ A. n e. Z ( `' ( F ` n ) " A ) = ( ( H ` n ) i^i dom ( F ` n ) ) ) -> E. h ( h : Z --> S /\ A. n e. Z ( `' ( F ` n ) " A ) = ( ( h ` n ) i^i dom ( F ` n ) ) ) )
60	25, 44, 59	syl2anc 693	1 |- ( ph -> E. h ( h : Z --> S /\ A. n e. Z ( `' ( F ` n ) " A ) = ( ( h ` n ) i^i dom ( F ` n ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   F/wnf 1708   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   i^i cin 3573   |-> cmpt 4729   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   -->wf 5884   ` cfv 5888

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896

This theorem is referenced by:  smfpimcc  41014