Theorem disjxp1 6204

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 6133	. . . . . . 7 |- ( y e. ( B X. C ) -> ( 1st ` y ) e. B )
2		xp1st 6133	. . . . . . 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 3960	. . . . . . . . . . . 12 |- (Disj x e. A B <-> A. z E* x e. A z e. B )
5	3, 4	sylib 121	. . . . . . . . . . 11 |- ( ph -> A. z E* x e. A z e. B )
6		1stexg 6135	. . . . . . . . . . . . 13 |- ( y e. _V -> ( 1st ` y ) e. _V )
7	6	elv 2730	. . . . . . . . . . . 12 |- ( 1st ` y ) e. _V
8		eleq1 2229	. . . . . . . . . . . . 13 |- ( z = ( 1st ` y ) -> ( z e. B <-> ( 1st ` y ) e. B ) )
9	8	rmobidv 2654	. . . . . . . . . . . 12 |- ( z = ( 1st ` y ) -> ( E* x e. A z e. B <-> E* x e. A ( 1st ` y ) e. B ) )
10	7, 9	spcv 2820	. . . . . . . . . . 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 2308	. . . . . . . . . . 11 |- F/_ x A
13		nfcv 2308	. . . . . . . . . . 11 |- F/_ w A
14		nfcsb1v 3078	. . . . . . . . . . . 12 |- F/_ x [_ w / x ]_ B
15	14	nfel2 2321	. . . . . . . . . . 11 |- F/ x ( 1st ` y ) e. [_ w / x ]_ B
16		csbeq1a 3054	. . . . . . . . . . . 12 |- ( x = w -> B = [_ w / x ]_ B )
17	16	eleq2d 2236	. . . . . . . . . . 11 |- ( x = w -> ( ( 1st ` y ) e. B <-> ( 1st ` y ) e. [_ w / x ]_ B ) )
18	12, 13, 15, 17	rmo4f 2924	. . . . . . . . . 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 121	. . . . . . . . 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 2554	. . . . . . . 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 2554	. . . . . . 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 406	. . . . . 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 2539	. . . . 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 2539	. . . 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 3078	. . . . . . 7 |- F/_ x [_ w / x ]_ C
26	14, 25	nfxp 4631	. . . . . 6 |- F/_ x ( [_ w / x ]_ B X. [_ w / x ]_ C )
27	26	nfel2 2321	. . . . 5 |- F/ x y e. ( [_ w / x ]_ B X. [_ w / x ]_ C )
28		csbeq1a 3054	. . . . . . 7 |- ( x = w -> C = [_ w / x ]_ C )
29	16, 28	xpeq12d 4629	. . . . . 6 |- ( x = w -> ( B X. C ) = ( [_ w / x ]_ B X. [_ w / x ]_ C ) )
30	29	eleq2d 2236	. . . . 5 |- ( x = w -> ( y e. ( B X. C ) <-> y e. ( [_ w / x ]_ B X. [_ w / x ]_ C ) ) )
31	12, 13, 27, 30	rmo4f 2924	. . . 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 133	. . 3 |- ( ph -> E* x e. A y e. ( B X. C ) )
33	32	alrimiv 1862	. 2 |- ( ph -> A. y E* x e. A y e. ( B X. C ) )
34		df-disj 3960	. 2 |- (Disj x e. A ( B X. C ) <-> A. y E* x e. A y e. ( B X. C ) )
35	33, 34	sylibr 133	1 |- ( ph -> Disj x e. A ( B X. C ) )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1341   = wceq 1343   e. wcel 2136   A.wral 2444   E*wrmo 2447   _Vcvv 2726   [_csb 3045  Disj wdisj 3959   X. cxp 4602   ` cfv 5188   1stc1st 6106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rmo 2452  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-disj 3960  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fo 5194  df-fv 5196  df-1st 6108

This theorem is referenced by:  disjsnxp  6205