Theorem disjxp1 6231

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 6160	. . . . . . 7 |- ( y e. ( B X. C ) -> ( 1st ` y ) e. B )
2		xp1st 6160	. . . . . . 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 3978	. . . . . . . . . . . 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 6162	. . . . . . . . . . . . 13 |- ( y e. _V -> ( 1st ` y ) e. _V )
7	6	elv 2741	. . . . . . . . . . . 12 |- ( 1st ` y ) e. _V
8		eleq1 2240	. . . . . . . . . . . . 13 |- ( z = ( 1st ` y ) -> ( z e. B <-> ( 1st ` y ) e. B ) )
9	8	rmobidv 2665	. . . . . . . . . . . 12 |- ( z = ( 1st ` y ) -> ( E* x e. A z e. B <-> E* x e. A ( 1st ` y ) e. B ) )
10	7, 9	spcv 2831	. . . . . . . . . . 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 2319	. . . . . . . . . . 11 |- F/_ x A
13		nfcv 2319	. . . . . . . . . . 11 |- F/_ w A
14		nfcsb1v 3090	. . . . . . . . . . . 12 |- F/_ x [_ w / x ]_ B
15	14	nfel2 2332	. . . . . . . . . . 11 |- F/ x ( 1st ` y ) e. [_ w / x ]_ B
16		csbeq1a 3066	. . . . . . . . . . . 12 |- ( x = w -> B = [_ w / x ]_ B )
17	16	eleq2d 2247	. . . . . . . . . . 11 |- ( x = w -> ( ( 1st ` y ) e. B <-> ( 1st ` y ) e. [_ w / x ]_ B ) )
18	12, 13, 15, 17	rmo4f 2935	. . . . . . . . . 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 2565	. . . . . . . 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 2565	. . . . . . 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 2550	. . . . 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 2550	. . . 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 3090	. . . . . . 7 |- F/_ x [_ w / x ]_ C
26	14, 25	nfxp 4650	. . . . . 6 |- F/_ x ( [_ w / x ]_ B X. [_ w / x ]_ C )
27	26	nfel2 2332	. . . . 5 |- F/ x y e. ( [_ w / x ]_ B X. [_ w / x ]_ C )
28		csbeq1a 3066	. . . . . . 7 |- ( x = w -> C = [_ w / x ]_ C )
29	16, 28	xpeq12d 4648	. . . . . 6 |- ( x = w -> ( B X. C ) = ( [_ w / x ]_ B X. [_ w / x ]_ C ) )
30	29	eleq2d 2247	. . . . 5 |- ( x = w -> ( y e. ( B X. C ) <-> y e. ( [_ w / x ]_ B X. [_ w / x ]_ C ) ) )
31	12, 13, 27, 30	rmo4f 2935	. . . 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 1874	. 2 |- ( ph -> A. y E* x e. A y e. ( B X. C ) )
34		df-disj 3978	. 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 1351   = wceq 1353   e. wcel 2148   A.wral 2455   E*wrmo 2458   _Vcvv 2737   [_csb 3057  Disj wdisj 3977   X. cxp 4621   ` cfv 5212   1stc1st 6133

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rmo 2463  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-disj 3978  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fo 5218  df-fv 5220  df-1st 6135

This theorem is referenced by:  disjsnxp  6232