Theorem disjf1 39369

Description: A 1 to 1 mapping built from disjoint, nonempty sets. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
disjf1.xph	|- F/ x ph
disjf1.f	|- F = ( x e. A |-> B )
disjf1.b	|- ( ( ph /\ x e. A ) -> B e. V )
disjf1.n0	|- ( ( ph /\ x e. A ) -> B =/= (/) )
disjf1.dj	|- ( ph -> Disj x e. A B )

Assertion

Ref	Expression
disjf1	|- ( ph -> F : A -1-1-> V )

Distinct variable groups:   x, A   x, V

Allowed substitution hints:   ph( x)   B( x)   F( x)

Proof of Theorem disjf1

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

Step	Hyp	Ref	Expression
1		disjf1.xph	. . . . . . 7 |- F/ x ph
2		nfv 1843	. . . . . . 7 |- F/ x y e. A
3	1, 2	nfan 1828	. . . . . 6 |- F/ x ( ph /\ y e. A )
4		nfcsb1v 3549	. . . . . . 7 |- F/_ x [_ y / x ]_ B
5		nfcv 2764	. . . . . . 7 |- F/_ x V
6	4, 5	nfel 2777	. . . . . 6 |- F/ x [_ y / x ]_ B e. V
7	3, 6	nfim 1825	. . . . 5 |- F/ x ( ( ph /\ y e. A ) -> [_ y / x ]_ B e. V )
8		eleq1 2689	. . . . . . 7 |- ( x = y -> ( x e. A <-> y e. A ) )
9	8	anbi2d 740	. . . . . 6 |- ( x = y -> ( ( ph /\ x e. A ) <-> ( ph /\ y e. A ) ) )
10		csbeq1a 3542	. . . . . . 7 |- ( x = y -> B = [_ y / x ]_ B )
11	10	eleq1d 2686	. . . . . 6 |- ( x = y -> ( B e. V <-> [_ y / x ]_ B e. V ) )
12	9, 11	imbi12d 334	. . . . 5 |- ( x = y -> ( ( ( ph /\ x e. A ) -> B e. V ) <-> ( ( ph /\ y e. A ) -> [_ y / x ]_ B e. V ) ) )
13		disjf1.b	. . . . 5 |- ( ( ph /\ x e. A ) -> B e. V )
14	7, 12, 13	chvar 2262	. . . 4 |- ( ( ph /\ y e. A ) -> [_ y / x ]_ B e. V )
15	14	ralrimiva 2966	. . 3 |- ( ph -> A. y e. A [_ y / x ]_ B e. V )
16		inidm 3822	. . . . . . . . 9 |- ( [_ y / x ]_ B i^i [_ y / x ]_ B ) = [_ y / x ]_ B
17	16	eqcomi 2631	. . . . . . . 8 |- [_ y / x ]_ B = ( [_ y / x ]_ B i^i [_ y / x ]_ B )
18	17	a1i 11	. . . . . . 7 |- ( ( ( ( ph /\ ( y e. A /\ z e. A ) ) /\ [_ y / x ]_ B = [_ z / x ]_ B ) /\ -. y = z ) -> [_ y / x ]_ B = ( [_ y / x ]_ B i^i [_ y / x ]_ B ) )
19		ineq2 3808	. . . . . . . 8 |- ( [_ y / x ]_ B = [_ z / x ]_ B -> ( [_ y / x ]_ B i^i [_ y / x ]_ B ) = ( [_ y / x ]_ B i^i [_ z / x ]_ B ) )
20	19	ad2antlr 763	. . . . . . 7 |- ( ( ( ( ph /\ ( y e. A /\ z e. A ) ) /\ [_ y / x ]_ B = [_ z / x ]_ B ) /\ -. y = z ) -> ( [_ y / x ]_ B i^i [_ y / x ]_ B ) = ( [_ y / x ]_ B i^i [_ z / x ]_ B ) )
21		disjf1.dj	. . . . . . . . . 10 |- ( ph -> Disj x e. A B )
22		nfcv 2764	. . . . . . . . . . 11 |- F/_ w B
23		nfcsb1v 3549	. . . . . . . . . . 11 |- F/_ x [_ w / x ]_ B
24		csbeq1a 3542	. . . . . . . . . . 11 |- ( x = w -> B = [_ w / x ]_ B )
25	22, 23, 24	cbvdisj 4630	. . . . . . . . . 10 |- (Disj x e. A B <-> Disj w e. A [_ w / x ]_ B )
26	21, 25	sylib 208	. . . . . . . . 9 |- ( ph -> Disj w e. A [_ w / x ]_ B )
27	26	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( ph /\ ( y e. A /\ z e. A ) ) /\ [_ y / x ]_ B = [_ z / x ]_ B ) /\ -. y = z ) -> Disj w e. A [_ w / x ]_ B )
28		simpllr 799	. . . . . . . 8 |- ( ( ( ( ph /\ ( y e. A /\ z e. A ) ) /\ [_ y / x ]_ B = [_ z / x ]_ B ) /\ -. y = z ) -> ( y e. A /\ z e. A ) )
29		neqne 2802	. . . . . . . . 9 |- ( -. y = z -> y =/= z )
30	29	adantl 482	. . . . . . . 8 |- ( ( ( ( ph /\ ( y e. A /\ z e. A ) ) /\ [_ y / x ]_ B = [_ z / x ]_ B ) /\ -. y = z ) -> y =/= z )
31		csbeq1 3536	. . . . . . . . 9 |- ( w = y -> [_ w / x ]_ B = [_ y / x ]_ B )
32		csbeq1 3536	. . . . . . . . 9 |- ( w = z -> [_ w / x ]_ B = [_ z / x ]_ B )
33	31, 32	disji2 4636	. . . . . . . 8 |- ( (Disj w e. A [_ w / x ]_ B /\ ( y e. A /\ z e. A ) /\ y =/= z ) -> ( [_ y / x ]_ B i^i [_ z / x ]_ B ) = (/) )
34	27, 28, 30, 33	syl3anc 1326	. . . . . . 7 |- ( ( ( ( ph /\ ( y e. A /\ z e. A ) ) /\ [_ y / x ]_ B = [_ z / x ]_ B ) /\ -. y = z ) -> ( [_ y / x ]_ B i^i [_ z / x ]_ B ) = (/) )
35	18, 20, 34	3eqtrd 2660	. . . . . 6 |- ( ( ( ( ph /\ ( y e. A /\ z e. A ) ) /\ [_ y / x ]_ B = [_ z / x ]_ B ) /\ -. y = z ) -> [_ y / x ]_ B = (/) )
36		nfcv 2764	. . . . . . . . . . . 12 |- F/_ x (/)
37	4, 36	nfne 2894	. . . . . . . . . . 11 |- F/ x [_ y / x ]_ B =/= (/)
38	3, 37	nfim 1825	. . . . . . . . . 10 |- F/ x ( ( ph /\ y e. A ) -> [_ y / x ]_ B =/= (/) )
39	10	neeq1d 2853	. . . . . . . . . . 11 |- ( x = y -> ( B =/= (/) <-> [_ y / x ]_ B =/= (/) ) )
40	9, 39	imbi12d 334	. . . . . . . . . 10 |- ( x = y -> ( ( ( ph /\ x e. A ) -> B =/= (/) ) <-> ( ( ph /\ y e. A ) -> [_ y / x ]_ B =/= (/) ) ) )
41		disjf1.n0	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> B =/= (/) )
42	38, 40, 41	chvar 2262	. . . . . . . . 9 |- ( ( ph /\ y e. A ) -> [_ y / x ]_ B =/= (/) )
43	42	adantrr 753	. . . . . . . 8 |- ( ( ph /\ ( y e. A /\ z e. A ) ) -> [_ y / x ]_ B =/= (/) )
44	43	ad2antrr 762	. . . . . . 7 |- ( ( ( ( ph /\ ( y e. A /\ z e. A ) ) /\ [_ y / x ]_ B = [_ z / x ]_ B ) /\ -. y = z ) -> [_ y / x ]_ B =/= (/) )
45	44	neneqd 2799	. . . . . 6 |- ( ( ( ( ph /\ ( y e. A /\ z e. A ) ) /\ [_ y / x ]_ B = [_ z / x ]_ B ) /\ -. y = z ) -> -. [_ y / x ]_ B = (/) )
46	35, 45	condan 835	. . . . 5 |- ( ( ( ph /\ ( y e. A /\ z e. A ) ) /\ [_ y / x ]_ B = [_ z / x ]_ B ) -> y = z )
47	46	ex 450	. . . 4 |- ( ( ph /\ ( y e. A /\ z e. A ) ) -> ( [_ y / x ]_ B = [_ z / x ]_ B -> y = z ) )
48	47	ralrimivva 2971	. . 3 |- ( ph -> A. y e. A A. z e. A ( [_ y / x ]_ B = [_ z / x ]_ B -> y = z ) )
49	15, 48	jca 554	. 2 |- ( ph -> ( A. y e. A [_ y / x ]_ B e. V /\ A. y e. A A. z e. A ( [_ y / x ]_ B = [_ z / x ]_ B -> y = z ) ) )
50		disjf1.f	. . . 4 |- F = ( x e. A |-> B )
51		nfcv 2764	. . . . 5 |- F/_ y B
52	51, 4, 10	cbvmpt 4749	. . . 4 |- ( x e. A |-> B ) = ( y e. A |-> [_ y / x ]_ B )
53	50, 52	eqtri 2644	. . 3 |- F = ( y e. A |-> [_ y / x ]_ B )
54		csbeq1 3536	. . 3 |- ( y = z -> [_ y / x ]_ B = [_ z / x ]_ B )
55	53, 54	f1mpt 6518	. 2 |- ( F : A -1-1-> V <-> ( A. y e. A [_ y / x ]_ B e. V /\ A. y e. A A. z e. A ( [_ y / x ]_ B = [_ z / x ]_ B -> y = z ) ) )
56	49, 55	sylibr 224	1 |- ( ph -> F : A -1-1-> V )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   =/= wne 2794   A.wral 2912   [_csb 3533   i^i cin 3573   (/)c0 3915  Disj wdisj 4620   |-> cmpt 4729   -1-1->wf1 5885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fv 5896

This theorem is referenced by:  disjf1o  39378  meadjiunlem  40682