Theorem disjrnmpt2 39375

Description: Disjointness of the range of a function in map-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypothesis

Ref	Expression
disjrnmpt2.1	|- F = ( x e. A |-> B )

Assertion

Ref	Expression
disjrnmpt2	|- (Disj x e. A B -> Disj y e. ran F y )

Distinct variable groups:   x, A   y, F

Allowed substitution hints:   A( y)   B( x, y)   F( x)

Proof of Theorem disjrnmpt2

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

Step	Hyp	Ref	Expression
1		id 22	. . . . . . 7 |- ( y = w -> y = w )
2	1	cbvdisjv 4631	. . . . . 6 |- (Disj y e. ran F y <-> Disj w e. ran F w )
3	2	notbii 310	. . . . 5 |- ( -. Disj y e. ran F y <-> -. Disj w e. ran F w )
4		id 22	. . . . . . 7 |- ( w = v -> w = v )
5	4	ndisj2 39218	. . . . . 6 |- ( -. Disj w e. ran F w <-> E. w e. ran F E. v e. ran F ( w =/= v /\ ( w i^i v ) =/= (/) ) )
6	5	biimpi 206	. . . . 5 |- ( -. Disj w e. ran F w -> E. w e. ran F E. v e. ran F ( w =/= v /\ ( w i^i v ) =/= (/) ) )
7	3, 6	sylbi 207	. . . 4 |- ( -. Disj y e. ran F y -> E. w e. ran F E. v e. ran F ( w =/= v /\ ( w i^i v ) =/= (/) ) )
8		disjrnmpt2.1	. . . . . . . . . . . . . 14 |- F = ( x e. A |-> B )
9	8	elrnmpt 5372	. . . . . . . . . . . . 13 |- ( w e. ran F -> ( w e. ran F <-> E. x e. A w = B ) )
10	9	ibi 256	. . . . . . . . . . . 12 |- ( w e. ran F -> E. x e. A w = B )
11	10	adantr 481	. . . . . . . . . . 11 |- ( ( w e. ran F /\ v e. ran F ) -> E. x e. A w = B )
12		nfcv 2764	. . . . . . . . . . . . . . . 16 |- F/_ z B
13		nfcsb1v 3549	. . . . . . . . . . . . . . . 16 |- F/_ x [_ z / x ]_ B
14		csbeq1a 3542	. . . . . . . . . . . . . . . 16 |- ( x = z -> B = [_ z / x ]_ B )
15	12, 13, 14	cbvmpt 4749	. . . . . . . . . . . . . . 15 |- ( x e. A |-> B ) = ( z e. A |-> [_ z / x ]_ B )
16	8, 15	eqtri 2644	. . . . . . . . . . . . . 14 |- F = ( z e. A |-> [_ z / x ]_ B )
17	16	elrnmpt 5372	. . . . . . . . . . . . 13 |- ( v e. ran F -> ( v e. ran F <-> E. z e. A v = [_ z / x ]_ B ) )
18	17	ibi 256	. . . . . . . . . . . 12 |- ( v e. ran F -> E. z e. A v = [_ z / x ]_ B )
19	18	adantl 482	. . . . . . . . . . 11 |- ( ( w e. ran F /\ v e. ran F ) -> E. z e. A v = [_ z / x ]_ B )
20	11, 19	jca 554	. . . . . . . . . 10 |- ( ( w e. ran F /\ v e. ran F ) -> ( E. x e. A w = B /\ E. z e. A v = [_ z / x ]_ B ) )
21		nfv 1843	. . . . . . . . . . 11 |- F/ z w = B
22		nfcv 2764	. . . . . . . . . . . 12 |- F/_ x v
23	22, 13	nfeq 2776	. . . . . . . . . . 11 |- F/ x v = [_ z / x ]_ B
24	21, 23	reean 3106	. . . . . . . . . 10 |- ( E. x e. A E. z e. A ( w = B /\ v = [_ z / x ]_ B ) <-> ( E. x e. A w = B /\ E. z e. A v = [_ z / x ]_ B ) )
25	20, 24	sylibr 224	. . . . . . . . 9 |- ( ( w e. ran F /\ v e. ran F ) -> E. x e. A E. z e. A ( w = B /\ v = [_ z / x ]_ B ) )
26	25	adantr 481	. . . . . . . 8 |- ( ( ( w e. ran F /\ v e. ran F ) /\ ( w =/= v /\ ( w i^i v ) =/= (/) ) ) -> E. x e. A E. z e. A ( w = B /\ v = [_ z / x ]_ B ) )
27		nfcv 2764	. . . . . . . . . . . 12 |- F/_ x w
28		nfmpt1 4747	. . . . . . . . . . . . . 14 |- F/_ x ( x e. A |-> B )
29	8, 28	nfcxfr 2762	. . . . . . . . . . . . 13 |- F/_ x F
30	29	nfrn 5368	. . . . . . . . . . . 12 |- F/_ x ran F
31	27, 30	nfel 2777	. . . . . . . . . . 11 |- F/ x w e. ran F
32	30	nfcri 2758	. . . . . . . . . . 11 |- F/ x v e. ran F
33	31, 32	nfan 1828	. . . . . . . . . 10 |- F/ x ( w e. ran F /\ v e. ran F )
34		nfv 1843	. . . . . . . . . 10 |- F/ x ( w =/= v /\ ( w i^i v ) =/= (/) )
35	33, 34	nfan 1828	. . . . . . . . 9 |- F/ x ( ( w e. ran F /\ v e. ran F ) /\ ( w =/= v /\ ( w i^i v ) =/= (/) ) )
36		simpll 790	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( w = B /\ v = [_ z / x ]_ B ) /\ x = z ) -> w = B )
37	14	adantl 482	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( w = B /\ v = [_ z / x ]_ B ) /\ x = z ) -> B = [_ z / x ]_ B )
38		id 22	. . . . . . . . . . . . . . . . . . . . 21 |- ( v = [_ z / x ]_ B -> v = [_ z / x ]_ B )
39	38	eqcomd 2628	. . . . . . . . . . . . . . . . . . . 20 |- ( v = [_ z / x ]_ B -> [_ z / x ]_ B = v )
40	39	ad2antlr 763	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( w = B /\ v = [_ z / x ]_ B ) /\ x = z ) -> [_ z / x ]_ B = v )
41	36, 37, 40	3eqtrd 2660	. . . . . . . . . . . . . . . . . 18 |- ( ( ( w = B /\ v = [_ z / x ]_ B ) /\ x = z ) -> w = v )
42	41	adantll 750	. . . . . . . . . . . . . . . . 17 |- ( ( ( w =/= v /\ ( w = B /\ v = [_ z / x ]_ B ) ) /\ x = z ) -> w = v )
43		simpll 790	. . . . . . . . . . . . . . . . . 18 |- ( ( ( w =/= v /\ ( w = B /\ v = [_ z / x ]_ B ) ) /\ x = z ) -> w =/= v )
44	43	neneqd 2799	. . . . . . . . . . . . . . . . 17 |- ( ( ( w =/= v /\ ( w = B /\ v = [_ z / x ]_ B ) ) /\ x = z ) -> -. w = v )
45	42, 44	pm2.65da 600	. . . . . . . . . . . . . . . 16 |- ( ( w =/= v /\ ( w = B /\ v = [_ z / x ]_ B ) ) -> -. x = z )
46	45	neqned 2801	. . . . . . . . . . . . . . 15 |- ( ( w =/= v /\ ( w = B /\ v = [_ z / x ]_ B ) ) -> x =/= z )
47	46	adantlr 751	. . . . . . . . . . . . . 14 |- ( ( ( w =/= v /\ ( w i^i v ) =/= (/) ) /\ ( w = B /\ v = [_ z / x ]_ B ) ) -> x =/= z )
48		id 22	. . . . . . . . . . . . . . . . . . 19 |- ( w = B -> w = B )
49	48	eqcomd 2628	. . . . . . . . . . . . . . . . . 18 |- ( w = B -> B = w )
50	49	ad2antrl 764	. . . . . . . . . . . . . . . . 17 |- ( ( ( w i^i v ) =/= (/) /\ ( w = B /\ v = [_ z / x ]_ B ) ) -> B = w )
51	39	ad2antll 765	. . . . . . . . . . . . . . . . 17 |- ( ( ( w i^i v ) =/= (/) /\ ( w = B /\ v = [_ z / x ]_ B ) ) -> [_ z / x ]_ B = v )
52	50, 51	ineq12d 3815	. . . . . . . . . . . . . . . 16 |- ( ( ( w i^i v ) =/= (/) /\ ( w = B /\ v = [_ z / x ]_ B ) ) -> ( B i^i [_ z / x ]_ B ) = ( w i^i v ) )
53		simpl 473	. . . . . . . . . . . . . . . 16 |- ( ( ( w i^i v ) =/= (/) /\ ( w = B /\ v = [_ z / x ]_ B ) ) -> ( w i^i v ) =/= (/) )
54	52, 53	eqnetrd 2861	. . . . . . . . . . . . . . 15 |- ( ( ( w i^i v ) =/= (/) /\ ( w = B /\ v = [_ z / x ]_ B ) ) -> ( B i^i [_ z / x ]_ B ) =/= (/) )
55	54	adantll 750	. . . . . . . . . . . . . 14 |- ( ( ( w =/= v /\ ( w i^i v ) =/= (/) ) /\ ( w = B /\ v = [_ z / x ]_ B ) ) -> ( B i^i [_ z / x ]_ B ) =/= (/) )
56	47, 55	jca 554	. . . . . . . . . . . . 13 |- ( ( ( w =/= v /\ ( w i^i v ) =/= (/) ) /\ ( w = B /\ v = [_ z / x ]_ B ) ) -> ( x =/= z /\ ( B i^i [_ z / x ]_ B ) =/= (/) ) )
57	56	ex 450	. . . . . . . . . . . 12 |- ( ( w =/= v /\ ( w i^i v ) =/= (/) ) -> ( ( w = B /\ v = [_ z / x ]_ B ) -> ( x =/= z /\ ( B i^i [_ z / x ]_ B ) =/= (/) ) ) )
58	57	adantl 482	. . . . . . . . . . 11 |- ( ( ( w e. ran F /\ v e. ran F ) /\ ( w =/= v /\ ( w i^i v ) =/= (/) ) ) -> ( ( w = B /\ v = [_ z / x ]_ B ) -> ( x =/= z /\ ( B i^i [_ z / x ]_ B ) =/= (/) ) ) )
59	58	reximdv 3016	. . . . . . . . . 10 |- ( ( ( w e. ran F /\ v e. ran F ) /\ ( w =/= v /\ ( w i^i v ) =/= (/) ) ) -> ( E. z e. A ( w = B /\ v = [_ z / x ]_ B ) -> E. z e. A ( x =/= z /\ ( B i^i [_ z / x ]_ B ) =/= (/) ) ) )
60	59	a1d 25	. . . . . . . . 9 |- ( ( ( w e. ran F /\ v e. ran F ) /\ ( w =/= v /\ ( w i^i v ) =/= (/) ) ) -> ( x e. A -> ( E. z e. A ( w = B /\ v = [_ z / x ]_ B ) -> E. z e. A ( x =/= z /\ ( B i^i [_ z / x ]_ B ) =/= (/) ) ) ) )
61	35, 60	reximdai 3012	. . . . . . . 8 |- ( ( ( w e. ran F /\ v e. ran F ) /\ ( w =/= v /\ ( w i^i v ) =/= (/) ) ) -> ( E. x e. A E. z e. A ( w = B /\ v = [_ z / x ]_ B ) -> E. x e. A E. z e. A ( x =/= z /\ ( B i^i [_ z / x ]_ B ) =/= (/) ) ) )
62	26, 61	mpd 15	. . . . . . 7 |- ( ( ( w e. ran F /\ v e. ran F ) /\ ( w =/= v /\ ( w i^i v ) =/= (/) ) ) -> E. x e. A E. z e. A ( x =/= z /\ ( B i^i [_ z / x ]_ B ) =/= (/) ) )
63	62	ex 450	. . . . . 6 |- ( ( w e. ran F /\ v e. ran F ) -> ( ( w =/= v /\ ( w i^i v ) =/= (/) ) -> E. x e. A E. z e. A ( x =/= z /\ ( B i^i [_ z / x ]_ B ) =/= (/) ) ) )
64	63	a1i 11	. . . . 5 |- ( -. Disj y e. ran F y -> ( ( w e. ran F /\ v e. ran F ) -> ( ( w =/= v /\ ( w i^i v ) =/= (/) ) -> E. x e. A E. z e. A ( x =/= z /\ ( B i^i [_ z / x ]_ B ) =/= (/) ) ) ) )
65	64	rexlimdvv 3037	. . . 4 |- ( -. Disj y e. ran F y -> ( E. w e. ran F E. v e. ran F ( w =/= v /\ ( w i^i v ) =/= (/) ) -> E. x e. A E. z e. A ( x =/= z /\ ( B i^i [_ z / x ]_ B ) =/= (/) ) ) )
66	7, 65	mpd 15	. . 3 |- ( -. Disj y e. ran F y -> E. x e. A E. z e. A ( x =/= z /\ ( B i^i [_ z / x ]_ B ) =/= (/) ) )
67		nfcv 2764	. . . . . 6 |- F/_ u B
68		nfcsb1v 3549	. . . . . 6 |- F/_ x [_ u / x ]_ B
69		csbeq1a 3542	. . . . . 6 |- ( x = u -> B = [_ u / x ]_ B )
70	67, 68, 69	cbvdisj 4630	. . . . 5 |- (Disj x e. A B <-> Disj u e. A [_ u / x ]_ B )
71	70	notbii 310	. . . 4 |- ( -. Disj x e. A B <-> -. Disj u e. A [_ u / x ]_ B )
72		csbeq1a 3542	. . . . . . 7 |- ( u = z -> [_ u / x ]_ B = [_ z / u ]_ [_ u / x ]_ B )
73		csbco 3543	. . . . . . . 8 |- [_ z / u ]_ [_ u / x ]_ B = [_ z / x ]_ B
74	73	a1i 11	. . . . . . 7 |- ( u = z -> [_ z / u ]_ [_ u / x ]_ B = [_ z / x ]_ B )
75	72, 74	eqtrd 2656	. . . . . 6 |- ( u = z -> [_ u / x ]_ B = [_ z / x ]_ B )
76	75	ndisj2 39218	. . . . 5 |- ( -. Disj u e. A [_ u / x ]_ B <-> E. u e. A E. z e. A ( u =/= z /\ ( [_ u / x ]_ B i^i [_ z / x ]_ B ) =/= (/) ) )
77		nfcv 2764	. . . . . . 7 |- F/_ x A
78		nfv 1843	. . . . . . . 8 |- F/ x u =/= z
79	68, 13	nfin 3820	. . . . . . . . 9 |- F/_ x ( [_ u / x ]_ B i^i [_ z / x ]_ B )
80		nfcv 2764	. . . . . . . . 9 |- F/_ x (/)
81	79, 80	nfne 2894	. . . . . . . 8 |- F/ x ( [_ u / x ]_ B i^i [_ z / x ]_ B ) =/= (/)
82	78, 81	nfan 1828	. . . . . . 7 |- F/ x ( u =/= z /\ ( [_ u / x ]_ B i^i [_ z / x ]_ B ) =/= (/) )
83	77, 82	nfrex 3007	. . . . . 6 |- F/ x E. z e. A ( u =/= z /\ ( [_ u / x ]_ B i^i [_ z / x ]_ B ) =/= (/) )
84		nfv 1843	. . . . . 6 |- F/ u E. z e. A ( x =/= z /\ ( B i^i [_ z / x ]_ B ) =/= (/) )
85		neeq1 2856	. . . . . . . 8 |- ( u = x -> ( u =/= z <-> x =/= z ) )
86		csbeq1 3536	. . . . . . . . . . 11 |- ( u = x -> [_ u / x ]_ B = [_ x / x ]_ B )
87		csbid 3541	. . . . . . . . . . . 12 |- [_ x / x ]_ B = B
88	87	a1i 11	. . . . . . . . . . 11 |- ( u = x -> [_ x / x ]_ B = B )
89	86, 88	eqtrd 2656	. . . . . . . . . 10 |- ( u = x -> [_ u / x ]_ B = B )
90	89	ineq1d 3813	. . . . . . . . 9 |- ( u = x -> ( [_ u / x ]_ B i^i [_ z / x ]_ B ) = ( B i^i [_ z / x ]_ B ) )
91	90	neeq1d 2853	. . . . . . . 8 |- ( u = x -> ( ( [_ u / x ]_ B i^i [_ z / x ]_ B ) =/= (/) <-> ( B i^i [_ z / x ]_ B ) =/= (/) ) )
92	85, 91	anbi12d 747	. . . . . . 7 |- ( u = x -> ( ( u =/= z /\ ( [_ u / x ]_ B i^i [_ z / x ]_ B ) =/= (/) ) <-> ( x =/= z /\ ( B i^i [_ z / x ]_ B ) =/= (/) ) ) )
93	92	rexbidv 3052	. . . . . 6 |- ( u = x -> ( E. z e. A ( u =/= z /\ ( [_ u / x ]_ B i^i [_ z / x ]_ B ) =/= (/) ) <-> E. z e. A ( x =/= z /\ ( B i^i [_ z / x ]_ B ) =/= (/) ) ) )
94	83, 84, 93	cbvrex 3168	. . . . 5 |- ( E. u e. A E. z e. A ( u =/= z /\ ( [_ u / x ]_ B i^i [_ z / x ]_ B ) =/= (/) ) <-> E. x e. A E. z e. A ( x =/= z /\ ( B i^i [_ z / x ]_ B ) =/= (/) ) )
95	76, 94	bitri 264	. . . 4 |- ( -. Disj u e. A [_ u / x ]_ B <-> E. x e. A E. z e. A ( x =/= z /\ ( B i^i [_ z / x ]_ B ) =/= (/) ) )
96	71, 95	bitri 264	. . 3 |- ( -. Disj x e. A B <-> E. x e. A E. z e. A ( x =/= z /\ ( B i^i [_ z / x ]_ B ) =/= (/) ) )
97	66, 96	sylibr 224	. 2 |- ( -. Disj y e. ran F y -> -. Disj x e. A B )
98	97	con4i 113	1 |- (Disj x e. A B -> Disj y e. ran F y )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   [_csb 3533   i^i cin 3573   (/)c0 3915  Disj wdisj 4620   |-> cmpt 4729   ran crn 5115

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  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-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-cnv 5122  df-dm 5124  df-rn 5125

This theorem is referenced by:  meadjiun  40683