Theorem disjabrexf 29396

Description: Rewriting a disjoint collection into a partition of its image set. (Contributed by Thierry Arnoux, 30-Dec-2016.) (Revised by Thierry Arnoux, 9-Mar-2017.)

Hypothesis

Ref	Expression
disjabrexf.1	|- F/_ x A

Assertion

Ref	Expression
disjabrexf	|- (Disj x e. A B -> Disj y e. { z | E. x e. A z = B } y )

Distinct variable groups:   x, y, z   y, A, z   y, B, z

Allowed substitution hints:   A( x)   B( x)

Proof of Theorem disjabrexf

Dummy variables i j are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfdisj1 4633	. . . 4 |- F/ xDisj x e. A B
2		nfcv 2764	. . . . 5 |- F/_ x y
3		disjabrexf.1	. . . . . . . . . . 11 |- F/_ x A
4	3	nfcri 2758	. . . . . . . . . 10 |- F/ x i e. A
5		nfcsb1v 3549	. . . . . . . . . . 11 |- F/_ x [_ i / x ]_ B
6	5	nfcri 2758	. . . . . . . . . 10 |- F/ x j e. [_ i / x ]_ B
7	4, 6	nfan 1828	. . . . . . . . 9 |- F/ x ( i e. A /\ j e. [_ i / x ]_ B )
8	7	nfab 2769	. . . . . . . 8 |- F/_ x { i | ( i e. A /\ j e. [_ i / x ]_ B ) }
9	8	nfuni 4442	. . . . . . 7 |- F/_ x U. { i | ( i e. A /\ j e. [_ i / x ]_ B ) }
10	9	nfcsb1 3548	. . . . . 6 |- F/_ x [_ U. { i | ( i e. A /\ j e. [_ i / x ]_ B ) } / x ]_ B
11	10	nfeq1 2778	. . . . 5 |- F/ x [_ U. { i | ( i e. A /\ j e. [_ i / x ]_ B ) } / x ]_ B = y
12	2, 11	nfral 2945	. . . 4 |- F/ x A. j e. y [_ U. { i | ( i e. A /\ j e. [_ i / x ]_ B ) } / x ]_ B = y
13		eqeq2 2633	. . . . 5 |- ( y = B -> ( [_ U. { i | ( i e. A /\ j e. [_ i / x ]_ B ) } / x ]_ B = y <-> [_ U. { i | ( i e. A /\ j e. [_ i / x ]_ B ) } / x ]_ B = B ) )
14	13	raleqbi1dv 3146	. . . 4 |- ( y = B -> ( A. j e. y [_ U. { i | ( i e. A /\ j e. [_ i / x ]_ B ) } / x ]_ B = y <-> A. j e. B [_ U. { i | ( i e. A /\ j e. [_ i / x ]_ B ) } / x ]_ B = B ) )
15		vex 3203	. . . . 5 |- y e. _V
16	15	a1i 11	. . . 4 |- (Disj x e. A B -> y e. _V )
17		simplll 798	. . . . . . . . . . . . 13 |- ( ( ( (Disj x e. A B /\ x e. A ) /\ j e. B ) /\ ( i e. A /\ j e. [_ i / x ]_ B ) ) -> Disj x e. A B )
18		simpllr 799	. . . . . . . . . . . . 13 |- ( ( ( (Disj x e. A B /\ x e. A ) /\ j e. B ) /\ ( i e. A /\ j e. [_ i / x ]_ B ) ) -> x e. A )
19		simprl 794	. . . . . . . . . . . . 13 |- ( ( ( (Disj x e. A B /\ x e. A ) /\ j e. B ) /\ ( i e. A /\ j e. [_ i / x ]_ B ) ) -> i e. A )
20		simplr 792	. . . . . . . . . . . . 13 |- ( ( ( (Disj x e. A B /\ x e. A ) /\ j e. B ) /\ ( i e. A /\ j e. [_ i / x ]_ B ) ) -> j e. B )
21		simprr 796	. . . . . . . . . . . . 13 |- ( ( ( (Disj x e. A B /\ x e. A ) /\ j e. B ) /\ ( i e. A /\ j e. [_ i / x ]_ B ) ) -> j e. [_ i / x ]_ B )
22		csbeq1a 3542	. . . . . . . . . . . . . 14 |- ( x = i -> B = [_ i / x ]_ B )
23	3, 5, 22	disjif2 29394	. . . . . . . . . . . . 13 |- ( (Disj x e. A B /\ ( x e. A /\ i e. A ) /\ ( j e. B /\ j e. [_ i / x ]_ B ) ) -> x = i )
24	17, 18, 19, 20, 21, 23	syl122anc 1335	. . . . . . . . . . . 12 |- ( ( ( (Disj x e. A B /\ x e. A ) /\ j e. B ) /\ ( i e. A /\ j e. [_ i / x ]_ B ) ) -> x = i )
25		simpr 477	. . . . . . . . . . . . . 14 |- ( ( ( (Disj x e. A B /\ x e. A ) /\ j e. B ) /\ x = i ) -> x = i )
26		simpllr 799	. . . . . . . . . . . . . 14 |- ( ( ( (Disj x e. A B /\ x e. A ) /\ j e. B ) /\ x = i ) -> x e. A )
27	25, 26	eqeltrrd 2702	. . . . . . . . . . . . 13 |- ( ( ( (Disj x e. A B /\ x e. A ) /\ j e. B ) /\ x = i ) -> i e. A )
28		simplr 792	. . . . . . . . . . . . . 14 |- ( ( ( (Disj x e. A B /\ x e. A ) /\ j e. B ) /\ x = i ) -> j e. B )
29	22	eleq2d 2687	. . . . . . . . . . . . . . 15 |- ( x = i -> ( j e. B <-> j e. [_ i / x ]_ B ) )
30	25, 29	syl 17	. . . . . . . . . . . . . 14 |- ( ( ( (Disj x e. A B /\ x e. A ) /\ j e. B ) /\ x = i ) -> ( j e. B <-> j e. [_ i / x ]_ B ) )
31	28, 30	mpbid 222	. . . . . . . . . . . . 13 |- ( ( ( (Disj x e. A B /\ x e. A ) /\ j e. B ) /\ x = i ) -> j e. [_ i / x ]_ B )
32	27, 31	jca 554	. . . . . . . . . . . 12 |- ( ( ( (Disj x e. A B /\ x e. A ) /\ j e. B ) /\ x = i ) -> ( i e. A /\ j e. [_ i / x ]_ B ) )
33	24, 32	impbida 877	. . . . . . . . . . 11 |- ( ( (Disj x e. A B /\ x e. A ) /\ j e. B ) -> ( ( i e. A /\ j e. [_ i / x ]_ B ) <-> x = i ) )
34		equcom 1945	. . . . . . . . . . 11 |- ( x = i <-> i = x )
35	33, 34	syl6bb 276	. . . . . . . . . 10 |- ( ( (Disj x e. A B /\ x e. A ) /\ j e. B ) -> ( ( i e. A /\ j e. [_ i / x ]_ B ) <-> i = x ) )
36	35	abbidv 2741	. . . . . . . . 9 |- ( ( (Disj x e. A B /\ x e. A ) /\ j e. B ) -> { i | ( i e. A /\ j e. [_ i / x ]_ B ) } = { i | i = x } )
37		df-sn 4178	. . . . . . . . 9 |- { x } = { i | i = x }
38	36, 37	syl6eqr 2674	. . . . . . . 8 |- ( ( (Disj x e. A B /\ x e. A ) /\ j e. B ) -> { i | ( i e. A /\ j e. [_ i / x ]_ B ) } = { x } )
39	38	unieqd 4446	. . . . . . 7 |- ( ( (Disj x e. A B /\ x e. A ) /\ j e. B ) -> U. { i | ( i e. A /\ j e. [_ i / x ]_ B ) } = U. { x } )
40		vex 3203	. . . . . . . 8 |- x e. _V
41	40	unisn 4451	. . . . . . 7 |- U. { x } = x
42	39, 41	syl6eq 2672	. . . . . 6 |- ( ( (Disj x e. A B /\ x e. A ) /\ j e. B ) -> U. { i | ( i e. A /\ j e. [_ i / x ]_ B ) } = x )
43		csbeq1 3536	. . . . . . 7 |- ( U. { i | ( i e. A /\ j e. [_ i / x ]_ B ) } = x -> [_ U. { i | ( i e. A /\ j e. [_ i / x ]_ B ) } / x ]_ B = [_ x / x ]_ B )
44		csbid 3541	. . . . . . 7 |- [_ x / x ]_ B = B
45	43, 44	syl6eq 2672	. . . . . 6 |- ( U. { i | ( i e. A /\ j e. [_ i / x ]_ B ) } = x -> [_ U. { i | ( i e. A /\ j e. [_ i / x ]_ B ) } / x ]_ B = B )
46	42, 45	syl 17	. . . . 5 |- ( ( (Disj x e. A B /\ x e. A ) /\ j e. B ) -> [_ U. { i | ( i e. A /\ j e. [_ i / x ]_ B ) } / x ]_ B = B )
47	46	ralrimiva 2966	. . . 4 |- ( (Disj x e. A B /\ x e. A ) -> A. j e. B [_ U. { i | ( i e. A /\ j e. [_ i / x ]_ B ) } / x ]_ B = B )
48	1, 12, 14, 16, 47	elabreximd 29348	. . 3 |- ( (Disj x e. A B /\ y e. { z | E. x e. A z = B } ) -> A. j e. y [_ U. { i | ( i e. A /\ j e. [_ i / x ]_ B ) } / x ]_ B = y )
49	48	ralrimiva 2966	. 2 |- (Disj x e. A B -> A. y e. { z | E. x e. A z = B } A. j e. y [_ U. { i | ( i e. A /\ j e. [_ i / x ]_ B ) } / x ]_ B = y )
50		invdisj 4638	. 2 |- ( A. y e. { z | E. x e. A z = B } A. j e. y [_ U. { i | ( i e. A /\ j e. [_ i / x ]_ B ) } / x ]_ B = y -> Disj y e. { z | E. x e. A z = B } y )
51	49, 50	syl 17	1 |- (Disj x e. A B -> Disj y e. { z | E. x e. A z = B } y )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   F/_wnfc 2751   A.wral 2912   E.wrex 2913   _Vcvv 3200   [_csb 3533   {csn 4177   U.cuni 4436  Disj wdisj 4620

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

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-rmo 2920  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-nul 3916  df-sn 4178  df-pr 4180  df-uni 4437  df-disj 4621

This theorem is referenced by:  measvunilem  30275