Theorem disjxp1 6215

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 6144	. . . . . . 7 |- ( y e. ( B X. C ) -> ( 1st ` y ) e. B )
2		xp1st 6144	. . . . . . 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 3967	. . . . . . . . . . . 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 6146	. . . . . . . . . . . . 13 |- ( y e. _V -> ( 1st ` y ) e. _V )
7	6	elv 2734	. . . . . . . . . . . 12 |- ( 1st ` y ) e. _V
8		eleq1 2233	. . . . . . . . . . . . 13 |- ( z = ( 1st ` y ) -> ( z e. B <-> ( 1st ` y ) e. B ) )
9	8	rmobidv 2658	. . . . . . . . . . . 12 |- ( z = ( 1st ` y ) -> ( E* x e. A z e. B <-> E* x e. A ( 1st ` y ) e. B ) )
10	7, 9	spcv 2824	. . . . . . . . . . 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 2312	. . . . . . . . . . 11 |- F/_ x A
13		nfcv 2312	. . . . . . . . . . 11 |- F/_ w A
14		nfcsb1v 3082	. . . . . . . . . . . 12 |- F/_ x [_ w / x ]_ B
15	14	nfel2 2325	. . . . . . . . . . 11 |- F/ x ( 1st ` y ) e. [_ w / x ]_ B
16		csbeq1a 3058	. . . . . . . . . . . 12 |- ( x = w -> B = [_ w / x ]_ B )
17	16	eleq2d 2240	. . . . . . . . . . 11 |- ( x = w -> ( ( 1st ` y ) e. B <-> ( 1st ` y ) e. [_ w / x ]_ B ) )
18	12, 13, 15, 17	rmo4f 2928	. . . . . . . . . 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 2558	. . . . . . . 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 2558	. . . . . . 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 2543	. . . . 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 2543	. . . 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 3082	. . . . . . 7 |- F/_ x [_ w / x ]_ C
26	14, 25	nfxp 4638	. . . . . 6 |- F/_ x ( [_ w / x ]_ B X. [_ w / x ]_ C )
27	26	nfel2 2325	. . . . 5 |- F/ x y e. ( [_ w / x ]_ B X. [_ w / x ]_ C )
28		csbeq1a 3058	. . . . . . 7 |- ( x = w -> C = [_ w / x ]_ C )
29	16, 28	xpeq12d 4636	. . . . . 6 |- ( x = w -> ( B X. C ) = ( [_ w / x ]_ B X. [_ w / x ]_ C ) )
30	29	eleq2d 2240	. . . . 5 |- ( x = w -> ( y e. ( B X. C ) <-> y e. ( [_ w / x ]_ B X. [_ w / x ]_ C ) ) )
31	12, 13, 27, 30	rmo4f 2928	. . . 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 1867	. 2 |- ( ph -> A. y E* x e. A y e. ( B X. C ) )
34		df-disj 3967	. 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 1346   = wceq 1348   e. wcel 2141   A.wral 2448   E*wrmo 2451   _Vcvv 2730   [_csb 3049  Disj wdisj 3966   X. cxp 4609   ` cfv 5198   1stc1st 6117

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rmo 2456  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-disj 3967  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fo 5204  df-fv 5206  df-1st 6119

This theorem is referenced by:  disjsnxp  6216