Theorem cantnfp1lem1 8575

Description: Lemma for cantnfp1 8578. (Contributed by Mario Carneiro, 20-Jun-2015.) (Revised by AV, 30-Jun-2019.)

Hypotheses

Ref	Expression
cantnfs.s	|- S = dom ( A CNF B )
cantnfs.a	|- ( ph -> A e. On )
cantnfs.b	|- ( ph -> B e. On )
cantnfp1.g	|- ( ph -> G e. S )
cantnfp1.x	|- ( ph -> X e. B )
cantnfp1.y	|- ( ph -> Y e. A )
cantnfp1.s	|- ( ph -> ( G supp (/) ) C_ X )
cantnfp1.f	|- F = ( t e. B |-> if ( t = X , Y , ( G ` t ) ) )

Assertion

Ref	Expression
cantnfp1lem1	|- ( ph -> F e. S )

Distinct variable groups:   t, B   t, A   t, S   t, G   ph, t   t, Y   t, X

Allowed substitution hint:   F( t)

Proof of Theorem cantnfp1lem1

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cantnfp1.y	. . . . 5 |- ( ph -> Y e. A )
2	1	adantr 481	. . . 4 |- ( ( ph /\ t e. B ) -> Y e. A )
3		cantnfp1.g	. . . . . . 7 |- ( ph -> G e. S )
4		cantnfs.s	. . . . . . . 8 |- S = dom ( A CNF B )
5		cantnfs.a	. . . . . . . 8 |- ( ph -> A e. On )
6		cantnfs.b	. . . . . . . 8 |- ( ph -> B e. On )
7	4, 5, 6	cantnfs 8563	. . . . . . 7 |- ( ph -> ( G e. S <-> ( G : B --> A /\ G finSupp (/) ) ) )
8	3, 7	mpbid 222	. . . . . 6 |- ( ph -> ( G : B --> A /\ G finSupp (/) ) )
9	8	simpld 475	. . . . 5 |- ( ph -> G : B --> A )
10	9	ffvelrnda 6359	. . . 4 |- ( ( ph /\ t e. B ) -> ( G ` t ) e. A )
11	2, 10	ifcld 4131	. . 3 |- ( ( ph /\ t e. B ) -> if ( t = X , Y , ( G ` t ) ) e. A )
12		cantnfp1.f	. . 3 |- F = ( t e. B |-> if ( t = X , Y , ( G ` t ) ) )
13	11, 12	fmptd 6385	. 2 |- ( ph -> F : B --> A )
14	8	simprd 479	. . . . . 6 |- ( ph -> G finSupp (/) )
15	14	fsuppimpd 8282	. . . . 5 |- ( ph -> ( G supp (/) ) e. Fin )
16		snfi 8038	. . . . 5 |- { X } e. Fin
17		unfi 8227	. . . . 5 |- ( ( ( G supp (/) ) e. Fin /\ { X } e. Fin ) -> ( ( G supp (/) ) u. { X } ) e. Fin )
18	15, 16, 17	sylancl 694	. . . 4 |- ( ph -> ( ( G supp (/) ) u. { X } ) e. Fin )
19		eldifi 3732	. . . . . . . 8 |- ( k e. ( B \ ( ( G supp (/) ) u. { X } ) ) -> k e. B )
20	19	adantl 482	. . . . . . 7 |- ( ( ph /\ k e. ( B \ ( ( G supp (/) ) u. { X } ) ) ) -> k e. B )
21	1	adantr 481	. . . . . . . 8 |- ( ( ph /\ k e. ( B \ ( ( G supp (/) ) u. { X } ) ) ) -> Y e. A )
22		fvex 6201	. . . . . . . 8 |- ( G ` k ) e. _V
23		ifexg 4157	. . . . . . . 8 |- ( ( Y e. A /\ ( G ` k ) e. _V ) -> if ( k = X , Y , ( G ` k ) ) e. _V )
24	21, 22, 23	sylancl 694	. . . . . . 7 |- ( ( ph /\ k e. ( B \ ( ( G supp (/) ) u. { X } ) ) ) -> if ( k = X , Y , ( G ` k ) ) e. _V )
25		eqeq1 2626	. . . . . . . . 9 |- ( t = k -> ( t = X <-> k = X ) )
26		fveq2 6191	. . . . . . . . 9 |- ( t = k -> ( G ` t ) = ( G ` k ) )
27	25, 26	ifbieq2d 4111	. . . . . . . 8 |- ( t = k -> if ( t = X , Y , ( G ` t ) ) = if ( k = X , Y , ( G ` k ) ) )
28	27, 12	fvmptg 6280	. . . . . . 7 |- ( ( k e. B /\ if ( k = X , Y , ( G ` k ) ) e. _V ) -> ( F ` k ) = if ( k = X , Y , ( G ` k ) ) )
29	20, 24, 28	syl2anc 693	. . . . . 6 |- ( ( ph /\ k e. ( B \ ( ( G supp (/) ) u. { X } ) ) ) -> ( F ` k ) = if ( k = X , Y , ( G ` k ) ) )
30		eldifn 3733	. . . . . . . . 9 |- ( k e. ( B \ ( ( G supp (/) ) u. { X } ) ) -> -. k e. ( ( G supp (/) ) u. { X } ) )
31	30	adantl 482	. . . . . . . 8 |- ( ( ph /\ k e. ( B \ ( ( G supp (/) ) u. { X } ) ) ) -> -. k e. ( ( G supp (/) ) u. { X } ) )
32		velsn 4193	. . . . . . . . 9 |- ( k e. { X } <-> k = X )
33		elun2 3781	. . . . . . . . 9 |- ( k e. { X } -> k e. ( ( G supp (/) ) u. { X } ) )
34	32, 33	sylbir 225	. . . . . . . 8 |- ( k = X -> k e. ( ( G supp (/) ) u. { X } ) )
35	31, 34	nsyl 135	. . . . . . 7 |- ( ( ph /\ k e. ( B \ ( ( G supp (/) ) u. { X } ) ) ) -> -. k = X )
36	35	iffalsed 4097	. . . . . 6 |- ( ( ph /\ k e. ( B \ ( ( G supp (/) ) u. { X } ) ) ) -> if ( k = X , Y , ( G ` k ) ) = ( G ` k ) )
37		ssun1 3776	. . . . . . . . 9 |- ( G supp (/) ) C_ ( ( G supp (/) ) u. { X } )
38		sscon 3744	. . . . . . . . 9 |- ( ( G supp (/) ) C_ ( ( G supp (/) ) u. { X } ) -> ( B \ ( ( G supp (/) ) u. { X } ) ) C_ ( B \ ( G supp (/) ) ) )
39	37, 38	ax-mp 5	. . . . . . . 8 |- ( B \ ( ( G supp (/) ) u. { X } ) ) C_ ( B \ ( G supp (/) ) )
40	39	sseli 3599	. . . . . . 7 |- ( k e. ( B \ ( ( G supp (/) ) u. { X } ) ) -> k e. ( B \ ( G supp (/) ) ) )
41		eqid 2622	. . . . . . . . 9 |- ( G supp (/) ) = ( G supp (/) )
42		eqimss2 3658	. . . . . . . . 9 |- ( ( G supp (/) ) = ( G supp (/) ) -> ( G supp (/) ) C_ ( G supp (/) ) )
43	41, 42	mp1i 13	. . . . . . . 8 |- ( ph -> ( G supp (/) ) C_ ( G supp (/) ) )
44		0ex 4790	. . . . . . . . 9 |- (/) e. _V
45	44	a1i 11	. . . . . . . 8 |- ( ph -> (/) e. _V )
46	9, 43, 6, 45	suppssr 7326	. . . . . . 7 |- ( ( ph /\ k e. ( B \ ( G supp (/) ) ) ) -> ( G ` k ) = (/) )
47	40, 46	sylan2 491	. . . . . 6 |- ( ( ph /\ k e. ( B \ ( ( G supp (/) ) u. { X } ) ) ) -> ( G ` k ) = (/) )
48	29, 36, 47	3eqtrd 2660	. . . . 5 |- ( ( ph /\ k e. ( B \ ( ( G supp (/) ) u. { X } ) ) ) -> ( F ` k ) = (/) )
49	13, 48	suppss 7325	. . . 4 |- ( ph -> ( F supp (/) ) C_ ( ( G supp (/) ) u. { X } ) )
50		ssfi 8180	. . . 4 |- ( ( ( ( G supp (/) ) u. { X } ) e. Fin /\ ( F supp (/) ) C_ ( ( G supp (/) ) u. { X } ) ) -> ( F supp (/) ) e. Fin )
51	18, 49, 50	syl2anc 693	. . 3 |- ( ph -> ( F supp (/) ) e. Fin )
52	12	funmpt2 5927	. . . . 5 |- Fun F
53	52	a1i 11	. . . 4 |- ( ph -> Fun F )
54		mptexg 6484	. . . . . 6 |- ( B e. On -> ( t e. B |-> if ( t = X , Y , ( G ` t ) ) ) e. _V )
55	12, 54	syl5eqel 2705	. . . . 5 |- ( B e. On -> F e. _V )
56	6, 55	syl 17	. . . 4 |- ( ph -> F e. _V )
57		funisfsupp 8280	. . . 4 |- ( ( Fun F /\ F e. _V /\ (/) e. _V ) -> ( F finSupp (/) <-> ( F supp (/) ) e. Fin ) )
58	53, 56, 45, 57	syl3anc 1326	. . 3 |- ( ph -> ( F finSupp (/) <-> ( F supp (/) ) e. Fin ) )
59	51, 58	mpbird 247	. 2 |- ( ph -> F finSupp (/) )
60	4, 5, 6	cantnfs 8563	. 2 |- ( ph -> ( F e. S <-> ( F : B --> A /\ F finSupp (/) ) ) )
61	13, 59, 60	mpbir2and 957	1 |- ( ph -> F e. S )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   u. cun 3572   C_ wss 3574   (/)c0 3915   ifcif 4086   {csn 4177   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   Oncon0 5723   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   supp csupp 7295   Fincfn 7955   finSupp cfsupp 8275   CNF ccnf 8558

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-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-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-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seqom 7543  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-fin 7959  df-fsupp 8276  df-cnf 8559

This theorem is referenced by:  cantnfp1lem2  8576  cantnfp1lem3  8577  cantnfp1  8578