Theorem disjxp1 6322

Description: The sets of a cartesian product are disjoint if the sets in the first argument are disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020.)

Hypothesis

Ref	Expression
disjxp1.1	|- ( ph -> Disj x e. A B )

Assertion

Ref	Expression
disjxp1	|- ( ph -> Disj x e. A ( B X. C ) )

Distinct variable groups:   x, A   ph, x

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

Proof of Theorem disjxp1

Dummy variables y z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xp1st 6251	. . . . . . 7 |- ( y e. ( B X. C ) -> ( 1st ` y ) e. B )
2		xp1st 6251	. . . . . . 7 |- ( y e. ( [_ w / x ]_ B X. [_ w / x ]_ C ) -> ( 1st ` y ) e. [_ w / x ]_ B )
3		disjxp1.1	. . . . . . . . . . . 12 |- ( ph -> Disj x e. A B )
4		df-disj 4022	. . . . . . . . . . . 12 |- (Disj x e. A B <-> A. z E* x e. A z e. B )
5	3, 4	sylib 122	. . . . . . . . . . 11 |- ( ph -> A. z E* x e. A z e. B )
6		1stexg 6253	. . . . . . . . . . . . 13 |- ( y e. _V -> ( 1st ` y ) e. _V )
7	6	elv 2776	. . . . . . . . . . . 12 |- ( 1st ` y ) e. _V
8		eleq1 2268	. . . . . . . . . . . . 13 |- ( z = ( 1st ` y ) -> ( z e. B <-> ( 1st ` y ) e. B ) )
9	8	rmobidv 2695	. . . . . . . . . . . 12 |- ( z = ( 1st ` y ) -> ( E* x e. A z e. B <-> E* x e. A ( 1st ` y ) e. B ) )
10	7, 9	spcv 2867	. . . . . . . . . . 11 |- ( A. z E* x e. A z e. B -> E* x e. A ( 1st ` y ) e. B )
11	5, 10	syl 14	. . . . . . . . . 10 |- ( ph -> E* x e. A ( 1st ` y ) e. B )
12		nfcv 2348	. . . . . . . . . . 11 |- F/_ x A
13		nfcv 2348	. . . . . . . . . . 11 |- F/_ w A
14		nfcsb1v 3126	. . . . . . . . . . . 12 |- F/_ x [_ w / x ]_ B
15	14	nfel2 2361	. . . . . . . . . . 11 |- F/ x ( 1st ` y ) e. [_ w / x ]_ B
16		csbeq1a 3102	. . . . . . . . . . . 12 |- ( x = w -> B = [_ w / x ]_ B )
17	16	eleq2d 2275	. . . . . . . . . . 11 |- ( x = w -> ( ( 1st ` y ) e. B <-> ( 1st ` y ) e. [_ w / x ]_ B ) )
18	12, 13, 15, 17	rmo4f 2971	. . . . . . . . . 10 |- ( E* x e. A ( 1st ` y ) e. B <-> A. x e. A A. w e. A ( ( ( 1st ` y ) e. B /\ ( 1st ` y ) e. [_ w / x ]_ B ) -> x = w ) )
19	11, 18	sylib 122	. . . . . . . . 9 |- ( ph -> A. x e. A A. w e. A ( ( ( 1st ` y ) e. B /\ ( 1st ` y ) e. [_ w / x ]_ B ) -> x = w ) )
20	19	r19.21bi 2594	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> A. w e. A ( ( ( 1st ` y ) e. B /\ ( 1st ` y ) e. [_ w / x ]_ B ) -> x = w ) )
21	20	r19.21bi 2594	. . . . . . 7 |- ( ( ( ph /\ x e. A ) /\ w e. A ) -> ( ( ( 1st ` y ) e. B /\ ( 1st ` y ) e. [_ w / x ]_ B ) -> x = w ) )
22	1, 2, 21	syl2ani 408	. . . . . 6 |- ( ( ( ph /\ x e. A ) /\ w e. A ) -> ( ( y e. ( B X. C ) /\ y e. ( [_ w / x ]_ B X. [_ w / x ]_ C ) ) -> x = w ) )
23	22	ralrimiva 2579	. . . . 5 |- ( ( ph /\ x e. A ) -> A. w e. A ( ( y e. ( B X. C ) /\ y e. ( [_ w / x ]_ B X. [_ w / x ]_ C ) ) -> x = w ) )
24	23	ralrimiva 2579	. . . 4 |- ( ph -> A. x e. A A. w e. A ( ( y e. ( B X. C ) /\ y e. ( [_ w / x ]_ B X. [_ w / x ]_ C ) ) -> x = w ) )
25		nfcsb1v 3126	. . . . . . 7 |- F/_ x [_ w / x ]_ C
26	14, 25	nfxp 4702	. . . . . 6 |- F/_ x ( [_ w / x ]_ B X. [_ w / x ]_ C )
27	26	nfel2 2361	. . . . 5 |- F/ x y e. ( [_ w / x ]_ B X. [_ w / x ]_ C )
28		csbeq1a 3102	. . . . . . 7 |- ( x = w -> C = [_ w / x ]_ C )
29	16, 28	xpeq12d 4700	. . . . . 6 |- ( x = w -> ( B X. C ) = ( [_ w / x ]_ B X. [_ w / x ]_ C ) )
30	29	eleq2d 2275	. . . . 5 |- ( x = w -> ( y e. ( B X. C ) <-> y e. ( [_ w / x ]_ B X. [_ w / x ]_ C ) ) )
31	12, 13, 27, 30	rmo4f 2971	. . . 4 |- ( E* x e. A y e. ( B X. C ) <-> A. x e. A A. w e. A ( ( y e. ( B X. C ) /\ y e. ( [_ w / x ]_ B X. [_ w / x ]_ C ) ) -> x = w ) )
32	24, 31	sylibr 134	. . 3 |- ( ph -> E* x e. A y e. ( B X. C ) )
33	32	alrimiv 1897	. 2 |- ( ph -> A. y E* x e. A y e. ( B X. C ) )
34		df-disj 4022	. 2 |- (Disj x e. A ( B X. C ) <-> A. y E* x e. A y e. ( B X. C ) )
35	33, 34	sylibr 134	1 |- ( ph -> Disj x e. A ( B X. C ) )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1371   = wceq 1373   e. wcel 2176   A.wral 2484   E*wrmo 2487   _Vcvv 2772   [_csb 3093  Disj wdisj 4021   X. cxp 4673   ` cfv 5271   1stc1st 6224

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rmo 2492  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-disj 4022  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-1st 6226

This theorem is referenced by:  disjsnxp  6323