Theorem aciunf1 29463

Description: Choice in an index union. (Contributed by Thierry Arnoux, 4-May-2020.)

Hypotheses

Ref	Expression
aciunf1.0	|- ( ph -> A e. V )
aciunf1.1	|- ( ( ph /\ j e. A ) -> B e. W )

Assertion

Ref	Expression
aciunf1	|- ( ph -> E. f ( f : U_ j e. A B -1-1-> U_ j e. A ( { j } X. B ) /\ A. k e. U_ j e. A B ( 2nd ` ( f ` k ) ) = k ) )

Distinct variable groups:   A, j, k, f   B, f, k   j, W   ph, f, j, k

Allowed substitution hints:   B( j)   V( f, j, k)   W( f, k)

Proof of Theorem aciunf1

Step	Hyp	Ref	Expression
1		ssrab2 3687	. . . 4 |- { j e. A | B =/= (/) } C_ A
2		aciunf1.0	. . . 4 |- ( ph -> A e. V )
3		ssexg 4804	. . . 4 |- ( ( { j e. A | B =/= (/) } C_ A /\ A e. V ) -> { j e. A | B =/= (/) } e. _V )
4	1, 2, 3	sylancr 695	. . 3 |- ( ph -> { j e. A | B =/= (/) } e. _V )
5		rabid 3116	. . . . . 6 |- ( j e. { j e. A | B =/= (/) } <-> ( j e. A /\ B =/= (/) ) )
6	5	biimpi 206	. . . . 5 |- ( j e. { j e. A | B =/= (/) } -> ( j e. A /\ B =/= (/) ) )
7	6	adantl 482	. . . 4 |- ( ( ph /\ j e. { j e. A | B =/= (/) } ) -> ( j e. A /\ B =/= (/) ) )
8	7	simprd 479	. . 3 |- ( ( ph /\ j e. { j e. A | B =/= (/) } ) -> B =/= (/) )
9		nfrab1 3122	. . 3 |- F/_ j { j e. A | B =/= (/) }
10	7	simpld 475	. . . 4 |- ( ( ph /\ j e. { j e. A | B =/= (/) } ) -> j e. A )
11		aciunf1.1	. . . 4 |- ( ( ph /\ j e. A ) -> B e. W )
12	10, 11	syldan 487	. . 3 |- ( ( ph /\ j e. { j e. A | B =/= (/) } ) -> B e. W )
13	4, 8, 9, 12	aciunf1lem 29462	. 2 |- ( ph -> E. f ( f : U_ j e. { j e. A | B =/= (/) } B -1-1-> U_ j e. { j e. A | B =/= (/) } ( { j } X. B ) /\ A. k e. U_ j e. { j e. A | B =/= (/) } B ( 2nd ` ( f ` k ) ) = k ) )
14		eqidd 2623	. . . . 5 |- ( ph -> f = f )
15		nfv 1843	. . . . . . 7 |- F/ j ph
16		nfcv 2764	. . . . . . . 8 |- F/_ j A
17		nfrab1 3122	. . . . . . . 8 |- F/_ j { j e. A | B = (/) }
18	16, 17	nfdif 3731	. . . . . . 7 |- F/_ j ( A \ { j e. A | B = (/) } )
19		difrab 3901	. . . . . . . . 9 |- ( { j e. A | T. } \ { j e. A | B = (/) } ) = { j e. A | ( T. /\ -. B = (/) ) }
20	16	rabtru 3361	. . . . . . . . . 10 |- { j e. A | T. } = A
21	20	difeq1i 3724	. . . . . . . . 9 |- ( { j e. A | T. } \ { j e. A | B = (/) } ) = ( A \ { j e. A | B = (/) } )
22		truan 1501	. . . . . . . . . . . . 13 |- ( ( T. /\ -. B = (/) ) <-> -. B = (/) )
23		df-ne 2795	. . . . . . . . . . . . 13 |- ( B =/= (/) <-> -. B = (/) )
24	22, 23	bitr4i 267	. . . . . . . . . . . 12 |- ( ( T. /\ -. B = (/) ) <-> B =/= (/) )
25	24	a1i 11	. . . . . . . . . . 11 |- ( T. -> ( ( T. /\ -. B = (/) ) <-> B =/= (/) ) )
26	25	rabbidv 3189	. . . . . . . . . 10 |- ( T. -> { j e. A | ( T. /\ -. B = (/) ) } = { j e. A | B =/= (/) } )
27	26	trud 1493	. . . . . . . . 9 |- { j e. A | ( T. /\ -. B = (/) ) } = { j e. A | B =/= (/) }
28	19, 21, 27	3eqtr3i 2652	. . . . . . . 8 |- ( A \ { j e. A | B = (/) } ) = { j e. A | B =/= (/) }
29	28	a1i 11	. . . . . . 7 |- ( ph -> ( A \ { j e. A | B = (/) } ) = { j e. A | B =/= (/) } )
30		eqidd 2623	. . . . . . 7 |- ( ph -> B = B )
31	15, 18, 9, 29, 30	iuneq12df 4544	. . . . . 6 |- ( ph -> U_ j e. ( A \ { j e. A | B = (/) } ) B = U_ j e. { j e. A | B =/= (/) } B )
32		rabid 3116	. . . . . . . . . . 11 |- ( j e. { j e. A | B = (/) } <-> ( j e. A /\ B = (/) ) )
33	32	biimpi 206	. . . . . . . . . 10 |- ( j e. { j e. A | B = (/) } -> ( j e. A /\ B = (/) ) )
34	33	adantl 482	. . . . . . . . 9 |- ( ( ph /\ j e. { j e. A | B = (/) } ) -> ( j e. A /\ B = (/) ) )
35	34	simprd 479	. . . . . . . 8 |- ( ( ph /\ j e. { j e. A | B = (/) } ) -> B = (/) )
36	35	ralrimiva 2966	. . . . . . 7 |- ( ph -> A. j e. { j e. A | B = (/) } B = (/) )
37	17	iunxdif3 4606	. . . . . . 7 |- ( A. j e. { j e. A | B = (/) } B = (/) -> U_ j e. ( A \ { j e. A | B = (/) } ) B = U_ j e. A B )
38	36, 37	syl 17	. . . . . 6 |- ( ph -> U_ j e. ( A \ { j e. A | B = (/) } ) B = U_ j e. A B )
39	31, 38	eqtr3d 2658	. . . . 5 |- ( ph -> U_ j e. { j e. A | B =/= (/) } B = U_ j e. A B )
40		eqidd 2623	. . . . . . 7 |- ( ph -> ( { j } X. B ) = ( { j } X. B ) )
41	15, 18, 9, 29, 40	iuneq12df 4544	. . . . . 6 |- ( ph -> U_ j e. ( A \ { j e. A | B = (/) } ) ( { j } X. B ) = U_ j e. { j e. A | B =/= (/) } ( { j } X. B ) )
42	35	xpeq2d 5139	. . . . . . . . 9 |- ( ( ph /\ j e. { j e. A | B = (/) } ) -> ( { j } X. B ) = ( { j } X. (/) ) )
43		xp0 5552	. . . . . . . . 9 |- ( { j } X. (/) ) = (/)
44	42, 43	syl6eq 2672	. . . . . . . 8 |- ( ( ph /\ j e. { j e. A | B = (/) } ) -> ( { j } X. B ) = (/) )
45	44	ralrimiva 2966	. . . . . . 7 |- ( ph -> A. j e. { j e. A | B = (/) } ( { j } X. B ) = (/) )
46	17	iunxdif3 4606	. . . . . . 7 |- ( A. j e. { j e. A | B = (/) } ( { j } X. B ) = (/) -> U_ j e. ( A \ { j e. A | B = (/) } ) ( { j } X. B ) = U_ j e. A ( { j } X. B ) )
47	45, 46	syl 17	. . . . . 6 |- ( ph -> U_ j e. ( A \ { j e. A | B = (/) } ) ( { j } X. B ) = U_ j e. A ( { j } X. B ) )
48	41, 47	eqtr3d 2658	. . . . 5 |- ( ph -> U_ j e. { j e. A | B =/= (/) } ( { j } X. B ) = U_ j e. A ( { j } X. B ) )
49	14, 39, 48	f1eq123d 6131	. . . 4 |- ( ph -> ( f : U_ j e. { j e. A | B =/= (/) } B -1-1-> U_ j e. { j e. A | B =/= (/) } ( { j } X. B ) <-> f : U_ j e. A B -1-1-> U_ j e. A ( { j } X. B ) ) )
50	39	raleqdv 3144	. . . 4 |- ( ph -> ( A. k e. U_ j e. { j e. A | B =/= (/) } B ( 2nd ` ( f ` k ) ) = k <-> A. k e. U_ j e. A B ( 2nd ` ( f ` k ) ) = k ) )
51	49, 50	anbi12d 747	. . 3 |- ( ph -> ( ( f : U_ j e. { j e. A | B =/= (/) } B -1-1-> U_ j e. { j e. A | B =/= (/) } ( { j } X. B ) /\ A. k e. U_ j e. { j e. A | B =/= (/) } B ( 2nd ` ( f ` k ) ) = k ) <-> ( f : U_ j e. A B -1-1-> U_ j e. A ( { j } X. B ) /\ A. k e. U_ j e. A B ( 2nd ` ( f ` k ) ) = k ) ) )
52	51	exbidv 1850	. 2 |- ( ph -> ( E. f ( f : U_ j e. { j e. A | B =/= (/) } B -1-1-> U_ j e. { j e. A | B =/= (/) } ( { j } X. B ) /\ A. k e. U_ j e. { j e. A | B =/= (/) } B ( 2nd ` ( f ` k ) ) = k ) <-> E. f ( f : U_ j e. A B -1-1-> U_ j e. A ( { j } X. B ) /\ A. k e. U_ j e. A B ( 2nd ` ( f ` k ) ) = k ) ) )
53	13, 52	mpbid 222	1 |- ( ph -> E. f ( f : U_ j e. A B -1-1-> U_ j e. A ( { j } X. B ) /\ A. k e. U_ j e. A B ( 2nd ` ( f ` k ) ) = k ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   T. wtru 1484   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   _Vcvv 3200   \ cdif 3571   C_ wss 3574   (/)c0 3915   {csn 4177   U_ciun 4520   X. cxp 5112   -1-1->wf1 5885   ` cfv 5888   2ndc2nd 7167

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-reg 8497  ax-inf2 8538  ax-ac2 9285

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-iin 4523  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-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-en 7956  df-r1 8627  df-rank 8628  df-card 8765  df-ac 8939

This theorem is referenced by:  fsumiunle  29575  esumiun  30156