Theorem pwfseqlem4a 9483

Description: Lemma for pwfseqlem4 9484. (Contributed by Mario Carneiro, 7-Jun-2016.)

Hypotheses

Ref	Expression
pwfseqlem4.g	|- ( ph -> G : ~P A -1-1-> U_ n e. om ( A ^m n ) )
pwfseqlem4.x	|- ( ph -> X C_ A )
pwfseqlem4.h	|- ( ph -> H : om -1-1-onto-> X )
pwfseqlem4.ps	|- ( ps <-> ( ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) /\ om ~<_ x ) )
pwfseqlem4.k	|- ( ( ph /\ ps ) -> K : U_ n e. om ( x ^m n ) -1-1-> x )
pwfseqlem4.d	|- D = ( G ` { w e. x | ( ( `' K ` w ) e. ran G /\ -. w e. ( `' G ` ( `' K ` w ) ) ) } )
pwfseqlem4.f	|- F = ( x e. _V , r e. _V |-> if ( x e. Fin , ( H ` ( card ` x ) ) , ( D ` |^| { z e. om | -. ( D ` z ) e. x } ) ) )

Assertion

Ref	Expression
pwfseqlem4a	|- ( ( ph /\ ( a C_ A /\ s C_ ( a X. a ) /\ s We a ) ) -> ( a F s ) e. A )

Distinct variable groups:   n, r, w, x, z   D, n, z   s, a, F   w, G   w, K   r, a, x, z, H, s   n, a, ph, s, r, x, z   ps, n, z   A, a, n, r, s, x, z

Allowed substitution hints:   ph( w)   ps( x, w, s, r, a)   A( w)   D( x, w, s, r, a)   F( x, z, w, n, r)   G( x, z, n, s, r, a)   H( w, n)   K( x, z, n, s, r, a)   X( x, z, w, n, s, r, a)

Proof of Theorem pwfseqlem4a

Step	Hyp	Ref	Expression
1		isfinite 8549	. . 3 |- ( a e. Fin <-> a ~< om )
2		simpr 477	. . . . . . 7 |- ( ( ph /\ a e. Fin ) -> a e. Fin )
3		vex 3203	. . . . . . 7 |- s e. _V
4		pwfseqlem4.g	. . . . . . . 8 |- ( ph -> G : ~P A -1-1-> U_ n e. om ( A ^m n ) )
5		pwfseqlem4.x	. . . . . . . 8 |- ( ph -> X C_ A )
6		pwfseqlem4.h	. . . . . . . 8 |- ( ph -> H : om -1-1-onto-> X )
7		pwfseqlem4.ps	. . . . . . . 8 |- ( ps <-> ( ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) /\ om ~<_ x ) )
8		pwfseqlem4.k	. . . . . . . 8 |- ( ( ph /\ ps ) -> K : U_ n e. om ( x ^m n ) -1-1-> x )
9		pwfseqlem4.d	. . . . . . . 8 |- D = ( G ` { w e. x | ( ( `' K ` w ) e. ran G /\ -. w e. ( `' G ` ( `' K ` w ) ) ) } )
10		pwfseqlem4.f	. . . . . . . 8 |- F = ( x e. _V , r e. _V |-> if ( x e. Fin , ( H ` ( card ` x ) ) , ( D ` |^| { z e. om | -. ( D ` z ) e. x } ) ) )
11	4, 5, 6, 7, 8, 9, 10	pwfseqlem2 9481	. . . . . . 7 |- ( ( a e. Fin /\ s e. _V ) -> ( a F s ) = ( H ` ( card ` a ) ) )
12	2, 3, 11	sylancl 694	. . . . . 6 |- ( ( ph /\ a e. Fin ) -> ( a F s ) = ( H ` ( card ` a ) ) )
13		f1of 6137	. . . . . . . . 9 |- ( H : om -1-1-onto-> X -> H : om --> X )
14	6, 13	syl 17	. . . . . . . 8 |- ( ph -> H : om --> X )
15	14, 5	fssd 6057	. . . . . . 7 |- ( ph -> H : om --> A )
16		ficardom 8787	. . . . . . 7 |- ( a e. Fin -> ( card ` a ) e. om )
17		ffvelrn 6357	. . . . . . 7 |- ( ( H : om --> A /\ ( card ` a ) e. om ) -> ( H ` ( card ` a ) ) e. A )
18	15, 16, 17	syl2an 494	. . . . . 6 |- ( ( ph /\ a e. Fin ) -> ( H ` ( card ` a ) ) e. A )
19	12, 18	eqeltrd 2701	. . . . 5 |- ( ( ph /\ a e. Fin ) -> ( a F s ) e. A )
20	19	ex 450	. . . 4 |- ( ph -> ( a e. Fin -> ( a F s ) e. A ) )
21	20	adantr 481	. . 3 |- ( ( ph /\ ( a C_ A /\ s C_ ( a X. a ) /\ s We a ) ) -> ( a e. Fin -> ( a F s ) e. A ) )
22	1, 21	syl5bir 233	. 2 |- ( ( ph /\ ( a C_ A /\ s C_ ( a X. a ) /\ s We a ) ) -> ( a ~< om -> ( a F s ) e. A ) )
23		omelon 8543	. . . . 5 |- om e. On
24		onenon 8775	. . . . 5 |- ( om e. On -> om e. dom card )
25	23, 24	ax-mp 5	. . . 4 |- om e. dom card
26		simpr3 1069	. . . . . 6 |- ( ( ph /\ ( a C_ A /\ s C_ ( a X. a ) /\ s We a ) ) -> s We a )
27		19.8a 2052	. . . . . 6 |- ( s We a -> E. s s We a )
28	26, 27	syl 17	. . . . 5 |- ( ( ph /\ ( a C_ A /\ s C_ ( a X. a ) /\ s We a ) ) -> E. s s We a )
29		ween 8858	. . . . 5 |- ( a e. dom card <-> E. s s We a )
30	28, 29	sylibr 224	. . . 4 |- ( ( ph /\ ( a C_ A /\ s C_ ( a X. a ) /\ s We a ) ) -> a e. dom card )
31		domtri2 8815	. . . 4 |- ( ( om e. dom card /\ a e. dom card ) -> ( om ~<_ a <-> -. a ~< om ) )
32	25, 30, 31	sylancr 695	. . 3 |- ( ( ph /\ ( a C_ A /\ s C_ ( a X. a ) /\ s We a ) ) -> ( om ~<_ a <-> -. a ~< om ) )
33		nfv 1843	. . . . . . 7 |- F/ r ( ph /\ ( ( a C_ A /\ s C_ ( a X. a ) /\ s We a ) /\ om ~<_ a ) )
34		nfcv 2764	. . . . . . . . 9 |- F/_ r a
35		nfmpt22 6723	. . . . . . . . . 10 |- F/_ r ( x e. _V , r e. _V |-> if ( x e. Fin , ( H ` ( card ` x ) ) , ( D ` |^| { z e. om | -. ( D ` z ) e. x } ) ) )
36	10, 35	nfcxfr 2762	. . . . . . . . 9 |- F/_ r F
37		nfcv 2764	. . . . . . . . 9 |- F/_ r s
38	34, 36, 37	nfov 6676	. . . . . . . 8 |- F/_ r ( a F s )
39	38	nfel1 2779	. . . . . . 7 |- F/ r ( a F s ) e. ( A \ a )
40	33, 39	nfim 1825	. . . . . 6 |- F/ r ( ( ph /\ ( ( a C_ A /\ s C_ ( a X. a ) /\ s We a ) /\ om ~<_ a ) ) -> ( a F s ) e. ( A \ a ) )
41		sseq1 3626	. . . . . . . . . 10 |- ( r = s -> ( r C_ ( a X. a ) <-> s C_ ( a X. a ) ) )
42		weeq1 5102	. . . . . . . . . 10 |- ( r = s -> ( r We a <-> s We a ) )
43	41, 42	3anbi23d 1402	. . . . . . . . 9 |- ( r = s -> ( ( a C_ A /\ r C_ ( a X. a ) /\ r We a ) <-> ( a C_ A /\ s C_ ( a X. a ) /\ s We a ) ) )
44	43	anbi1d 741	. . . . . . . 8 |- ( r = s -> ( ( ( a C_ A /\ r C_ ( a X. a ) /\ r We a ) /\ om ~<_ a ) <-> ( ( a C_ A /\ s C_ ( a X. a ) /\ s We a ) /\ om ~<_ a ) ) )
45	44	anbi2d 740	. . . . . . 7 |- ( r = s -> ( ( ph /\ ( ( a C_ A /\ r C_ ( a X. a ) /\ r We a ) /\ om ~<_ a ) ) <-> ( ph /\ ( ( a C_ A /\ s C_ ( a X. a ) /\ s We a ) /\ om ~<_ a ) ) ) )
46		oveq2 6658	. . . . . . . 8 |- ( r = s -> ( a F r ) = ( a F s ) )
47	46	eleq1d 2686	. . . . . . 7 |- ( r = s -> ( ( a F r ) e. ( A \ a ) <-> ( a F s ) e. ( A \ a ) ) )
48	45, 47	imbi12d 334	. . . . . 6 |- ( r = s -> ( ( ( ph /\ ( ( a C_ A /\ r C_ ( a X. a ) /\ r We a ) /\ om ~<_ a ) ) -> ( a F r ) e. ( A \ a ) ) <-> ( ( ph /\ ( ( a C_ A /\ s C_ ( a X. a ) /\ s We a ) /\ om ~<_ a ) ) -> ( a F s ) e. ( A \ a ) ) ) )
49		nfv 1843	. . . . . . . 8 |- F/ x ( ph /\ ( ( a C_ A /\ r C_ ( a X. a ) /\ r We a ) /\ om ~<_ a ) )
50		nfcv 2764	. . . . . . . . . 10 |- F/_ x a
51		nfmpt21 6722	. . . . . . . . . . 11 |- F/_ x ( x e. _V , r e. _V |-> if ( x e. Fin , ( H ` ( card ` x ) ) , ( D ` |^| { z e. om | -. ( D ` z ) e. x } ) ) )
52	10, 51	nfcxfr 2762	. . . . . . . . . 10 |- F/_ x F
53		nfcv 2764	. . . . . . . . . 10 |- F/_ x r
54	50, 52, 53	nfov 6676	. . . . . . . . 9 |- F/_ x ( a F r )
55	54	nfel1 2779	. . . . . . . 8 |- F/ x ( a F r ) e. ( A \ a )
56	49, 55	nfim 1825	. . . . . . 7 |- F/ x ( ( ph /\ ( ( a C_ A /\ r C_ ( a X. a ) /\ r We a ) /\ om ~<_ a ) ) -> ( a F r ) e. ( A \ a ) )
57		sseq1 3626	. . . . . . . . . . . 12 |- ( x = a -> ( x C_ A <-> a C_ A ) )
58		xpeq12 5134	. . . . . . . . . . . . . 14 |- ( ( x = a /\ x = a ) -> ( x X. x ) = ( a X. a ) )
59	58	anidms 677	. . . . . . . . . . . . 13 |- ( x = a -> ( x X. x ) = ( a X. a ) )
60	59	sseq2d 3633	. . . . . . . . . . . 12 |- ( x = a -> ( r C_ ( x X. x ) <-> r C_ ( a X. a ) ) )
61		weeq2 5103	. . . . . . . . . . . 12 |- ( x = a -> ( r We x <-> r We a ) )
62	57, 60, 61	3anbi123d 1399	. . . . . . . . . . 11 |- ( x = a -> ( ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) <-> ( a C_ A /\ r C_ ( a X. a ) /\ r We a ) ) )
63		breq2 4657	. . . . . . . . . . 11 |- ( x = a -> ( om ~<_ x <-> om ~<_ a ) )
64	62, 63	anbi12d 747	. . . . . . . . . 10 |- ( x = a -> ( ( ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) /\ om ~<_ x ) <-> ( ( a C_ A /\ r C_ ( a X. a ) /\ r We a ) /\ om ~<_ a ) ) )
65	7, 64	syl5bb 272	. . . . . . . . 9 |- ( x = a -> ( ps <-> ( ( a C_ A /\ r C_ ( a X. a ) /\ r We a ) /\ om ~<_ a ) ) )
66	65	anbi2d 740	. . . . . . . 8 |- ( x = a -> ( ( ph /\ ps ) <-> ( ph /\ ( ( a C_ A /\ r C_ ( a X. a ) /\ r We a ) /\ om ~<_ a ) ) ) )
67		oveq1 6657	. . . . . . . . 9 |- ( x = a -> ( x F r ) = ( a F r ) )
68		difeq2 3722	. . . . . . . . 9 |- ( x = a -> ( A \ x ) = ( A \ a ) )
69	67, 68	eleq12d 2695	. . . . . . . 8 |- ( x = a -> ( ( x F r ) e. ( A \ x ) <-> ( a F r ) e. ( A \ a ) ) )
70	66, 69	imbi12d 334	. . . . . . 7 |- ( x = a -> ( ( ( ph /\ ps ) -> ( x F r ) e. ( A \ x ) ) <-> ( ( ph /\ ( ( a C_ A /\ r C_ ( a X. a ) /\ r We a ) /\ om ~<_ a ) ) -> ( a F r ) e. ( A \ a ) ) ) )
71	4, 5, 6, 7, 8, 9, 10	pwfseqlem3 9482	. . . . . . 7 |- ( ( ph /\ ps ) -> ( x F r ) e. ( A \ x ) )
72	56, 70, 71	chvar 2262	. . . . . 6 |- ( ( ph /\ ( ( a C_ A /\ r C_ ( a X. a ) /\ r We a ) /\ om ~<_ a ) ) -> ( a F r ) e. ( A \ a ) )
73	40, 48, 72	chvar 2262	. . . . 5 |- ( ( ph /\ ( ( a C_ A /\ s C_ ( a X. a ) /\ s We a ) /\ om ~<_ a ) ) -> ( a F s ) e. ( A \ a ) )
74	73	eldifad 3586	. . . 4 |- ( ( ph /\ ( ( a C_ A /\ s C_ ( a X. a ) /\ s We a ) /\ om ~<_ a ) ) -> ( a F s ) e. A )
75	74	expr 643	. . 3 |- ( ( ph /\ ( a C_ A /\ s C_ ( a X. a ) /\ s We a ) ) -> ( om ~<_ a -> ( a F s ) e. A ) )
76	32, 75	sylbird 250	. 2 |- ( ( ph /\ ( a C_ A /\ s C_ ( a X. a ) /\ s We a ) ) -> ( -. a ~< om -> ( a F s ) e. A ) )
77	22, 76	pm2.61d 170	1 |- ( ( ph /\ ( a C_ A /\ s C_ ( a X. a ) /\ s We a ) ) -> ( a F s ) e. A )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {crab 2916   _Vcvv 3200   \ cdif 3571   C_ wss 3574   ifcif 4086   ~Pcpw 4158   |^|cint 4475   U_ciun 4520   class class class wbr 4653   We wwe 5072   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   Oncon0 5723   -->wf 5884   -1-1->wf1 5885   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   omcom 7065   ^m cmap 7857   ~<_ cdom 7953   ~< csdm 7954   Fincfn 7955   cardccrd 8761

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  ax-un 6949  ax-inf2 8538

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-rmo 2920  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-int 4476  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-se 5074  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765

This theorem is referenced by:  pwfseqlem4  9484