Theorem disjxp1 6001

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 5936	. . . . . . 7 |- ( y e. ( B X. C ) -> ( 1st ` y ) e. B )
2		xp1st 5936	. . . . . . 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 3823	. . . . . . . . . . . 12 |- (Disj x e. A B <-> A. z E* x e. A z e. B )
5	3, 4	sylib 120	. . . . . . . . . . 11 |- ( ph -> A. z E* x e. A z e. B )
6		1stexg 5938	. . . . . . . . . . . . 13 |- ( y e. _V -> ( 1st ` y ) e. _V )
7	6	elv 2623	. . . . . . . . . . . 12 |- ( 1st ` y ) e. _V
8		eleq1 2150	. . . . . . . . . . . . 13 |- ( z = ( 1st ` y ) -> ( z e. B <-> ( 1st ` y ) e. B ) )
9	8	rmobidv 2555	. . . . . . . . . . . 12 |- ( z = ( 1st ` y ) -> ( E* x e. A z e. B <-> E* x e. A ( 1st ` y ) e. B ) )
10	7, 9	spcv 2712	. . . . . . . . . . 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 2228	. . . . . . . . . . 11 |- F/_ x A
13		nfcv 2228	. . . . . . . . . . 11 |- F/_ w A
14		nfcsb1v 2963	. . . . . . . . . . . 12 |- F/_ x [_ w / x ]_ B
15	14	nfel2 2241	. . . . . . . . . . 11 |- F/ x ( 1st ` y ) e. [_ w / x ]_ B
16		csbeq1a 2941	. . . . . . . . . . . 12 |- ( x = w -> B = [_ w / x ]_ B )
17	16	eleq2d 2157	. . . . . . . . . . 11 |- ( x = w -> ( ( 1st ` y ) e. B <-> ( 1st ` y ) e. [_ w / x ]_ B ) )
18	12, 13, 15, 17	rmo4f 2813	. . . . . . . . . 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 120	. . . . . . . . 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 2461	. . . . . . . 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 2461	. . . . . . 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 400	. . . . . 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 2446	. . . . 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 2446	. . . 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 2963	. . . . . . 7 |- F/_ x [_ w / x ]_ C
26	14, 25	nfxp 4464	. . . . . 6 |- F/_ x ( [_ w / x ]_ B X. [_ w / x ]_ C )
27	26	nfel2 2241	. . . . 5 |- F/ x y e. ( [_ w / x ]_ B X. [_ w / x ]_ C )
28		csbeq1a 2941	. . . . . . 7 |- ( x = w -> C = [_ w / x ]_ C )
29	16, 28	xpeq12d 4463	. . . . . 6 |- ( x = w -> ( B X. C ) = ( [_ w / x ]_ B X. [_ w / x ]_ C ) )
30	29	eleq2d 2157	. . . . 5 |- ( x = w -> ( y e. ( B X. C ) <-> y e. ( [_ w / x ]_ B X. [_ w / x ]_ C ) ) )
31	12, 13, 27, 30	rmo4f 2813	. . . 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 132	. . 3 |- ( ph -> E* x e. A y e. ( B X. C ) )
33	32	alrimiv 1802	. 2 |- ( ph -> A. y E* x e. A y e. ( B X. C ) )
34		df-disj 3823	. 2 |- (Disj x e. A ( B X. C ) <-> A. y E* x e. A y e. ( B X. C ) )
35	33, 34	sylibr 132	1 |- ( ph -> Disj x e. A ( B X. C ) )

Syntax hints:   -> wi 4   /\ wa 102   A.wal 1287   = wceq 1289   e. wcel 1438   A.wral 2359   E*wrmo 2362   _Vcvv 2619   [_csb 2933  Disj wdisj 3822   X. cxp 4436   ` cfv 5015   1stc1st 5909

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rmo 2367  df-v 2621  df-sbc 2841  df-csb 2934  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-disj 3823  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-fo 5021  df-fv 5023  df-1st 5911

This theorem is referenced by:  disjsnxp  6002