Theorem disjrdx 29404

Description: Re-index a disjunct collection statement. (Contributed by Thierry Arnoux, 7-Apr-2017.)

Hypotheses

Ref	Expression
disjrdx.1	|- ( ph -> F : A -1-1-onto-> C )
disjrdx.2	|- ( ( ph /\ y = ( F ` x ) ) -> D = B )

Assertion

Ref	Expression
disjrdx	|- ( ph -> (Disj x e. A B <-> Disj y e. C D ) )

Distinct variable groups:   x, y, A   y, B   x, C, y   x, D   x, F, y   ph, x, y

Allowed substitution hints:   B( x)   D( y)

Proof of Theorem disjrdx

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		disjrdx.1	. . . . . . 7 |- ( ph -> F : A -1-1-onto-> C )
2		f1of 6137	. . . . . . 7 |- ( F : A -1-1-onto-> C -> F : A --> C )
3	1, 2	syl 17	. . . . . 6 |- ( ph -> F : A --> C )
4	3	ffvelrnda 6359	. . . . 5 |- ( ( ph /\ x e. A ) -> ( F ` x ) e. C )
5		f1ofveu 6645	. . . . . . 7 |- ( ( F : A -1-1-onto-> C /\ y e. C ) -> E! x e. A ( F ` x ) = y )
6	1, 5	sylan 488	. . . . . 6 |- ( ( ph /\ y e. C ) -> E! x e. A ( F ` x ) = y )
7		eqcom 2629	. . . . . . 7 |- ( ( F ` x ) = y <-> y = ( F ` x ) )
8	7	reubii 3128	. . . . . 6 |- ( E! x e. A ( F ` x ) = y <-> E! x e. A y = ( F ` x ) )
9	6, 8	sylib 208	. . . . 5 |- ( ( ph /\ y e. C ) -> E! x e. A y = ( F ` x ) )
10		disjrdx.2	. . . . . 6 |- ( ( ph /\ y = ( F ` x ) ) -> D = B )
11	10	eleq2d 2687	. . . . 5 |- ( ( ph /\ y = ( F ` x ) ) -> ( z e. D <-> z e. B ) )
12	4, 9, 11	rmoxfrd 29333	. . . 4 |- ( ph -> ( E* y e. C z e. D <-> E* x e. A z e. B ) )
13	12	bicomd 213	. . 3 |- ( ph -> ( E* x e. A z e. B <-> E* y e. C z e. D ) )
14	13	albidv 1849	. 2 |- ( ph -> ( A. z E* x e. A z e. B <-> A. z E* y e. C z e. D ) )
15		df-disj 4621	. 2 |- (Disj x e. A B <-> A. z E* x e. A z e. B )
16		df-disj 4621	. 2 |- (Disj y e. C D <-> A. z E* y e. C z e. D )
17	14, 15, 16	3bitr4g 303	1 |- ( ph -> (Disj x e. A B <-> Disj y e. C D ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   E!wreu 2914   E*wrmo 2915  Disj wdisj 4620   -->wf 5884   -1-1-onto->wf1o 5887   ` 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-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-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  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-disj 4621  df-br 4654  df-opab 4713  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:  volmeas  30294  carsggect  30380