Theorem disjrnmpt2 37473

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 4387	. . . . . 6 |- (Disj y e. ran F y <-> Disj w e. ran F w )
3	2	notbii 298	. . . . 5 |- ( -. Disj y e. ran F y <-> -. Disj w e. ran F w )
4		id 22	. . . . . . 7 |- ( w = v -> w = v )
5	4	ndisj2 37399	. . . . . 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 198	. . . . 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 199	. . . 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 5084	. . . . . . . . . . . . 13 |- ( w e. ran F -> ( w e. ran F <-> E. x e. A w = B ) )
10	9	ibi 245	. . . . . . . . . . . 12 |- ( w e. ran F -> E. x e. A w = B )
11	10	adantr 467	. . . . . . . . . . 11 |- ( ( w e. ran F /\ v e. ran F ) -> E. x e. A w = B )
12		nfcv 2594	. . . . . . . . . . . . . . . 16 |- F/_ z B
13		nfcsb1v 3381	. . . . . . . . . . . . . . . 16 |- F/_ x [_ z / x ]_ B
14		csbeq1a 3374	. . . . . . . . . . . . . . . 16 |- ( x = z -> B = [_ z / x ]_ B )
15	12, 13, 14	cbvmpt 4497	. . . . . . . . . . . . . . 15 |- ( x e. A |-> B ) = ( z e. A |-> [_ z / x ]_ B )
16	8, 15	eqtri 2475	. . . . . . . . . . . . . 14 |- F = ( z e. A |-> [_ z / x ]_ B )
17	16	elrnmpt 5084	. . . . . . . . . . . . 13 |- ( v e. ran F -> ( v e. ran F <-> E. z e. A v = [_ z / x ]_ B ) )
18	17	ibi 245	. . . . . . . . . . . 12 |- ( v e. ran F -> E. z e. A v = [_ z / x ]_ B )
19	18	adantl 468	. . . . . . . . . . 11 |- ( ( w e. ran F /\ v e. ran F ) -> E. z e. A v = [_ z / x ]_ B )
20	11, 19	jca 535	. . . . . . . . . 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 1763	. . . . . . . . . . 11 |- F/ z w = B
22		nfcv 2594	. . . . . . . . . . . 12 |- F/_ x v
23	22, 13	nfeq 2605	. . . . . . . . . . 11 |- F/ x v = [_ z / x ]_ B
24	21, 23	reean 2959	. . . . . . . . . 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 216	. . . . . . . . 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 467	. . . . . . . 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 2594	. . . . . . . . . . . 12 |- F/_ x w
28		nfmpt1 4495	. . . . . . . . . . . . . 14 |- F/_ x ( x e. A |-> B )
29	8, 28	nfcxfr 2592	. . . . . . . . . . . . 13 |- F/_ x F
30	29	nfrn 5080	. . . . . . . . . . . 12 |- F/_ x ran F
31	27, 30	nfel 2606	. . . . . . . . . . 11 |- F/ x w e. ran F
32	30	nfcri 2588	. . . . . . . . . . 11 |- F/ x v e. ran F
33	31, 32	nfan 2013	. . . . . . . . . 10 |- F/ x ( w e. ran F /\ v e. ran F )
34		nfv 1763	. . . . . . . . . 10 |- F/ x ( w =/= v /\ ( w i^i v ) =/= (/) )
35	33, 34	nfan 2013	. . . . . . . . 9 |- F/ x ( ( w e. ran F /\ v e. ran F ) /\ ( w =/= v /\ ( w i^i v ) =/= (/) ) )
36		simpll 761	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( w = B /\ v = [_ z / x ]_ B ) /\ x = z ) -> w = B )
37	14	adantl 468	. . . . . . . . . . . . . . . . . . 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 2459	. . . . . . . . . . . . . . . . . . . 20 |- ( v = [_ z / x ]_ B -> [_ z / x ]_ B = v )
40	39	ad2antlr 734	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( w = B /\ v = [_ z / x ]_ B ) /\ x = z ) -> [_ z / x ]_ B = v )
41	36, 37, 40	3eqtrd 2491	. . . . . . . . . . . . . . . . . 18 |- ( ( ( w = B /\ v = [_ z / x ]_ B ) /\ x = z ) -> w = v )
42	41	adantll 721	. . . . . . . . . . . . . . . . 17 |- ( ( ( w =/= v /\ ( w = B /\ v = [_ z / x ]_ B ) ) /\ x = z ) -> w = v )
43		simpll 761	. . . . . . . . . . . . . . . . . 18 |- ( ( ( w =/= v /\ ( w = B /\ v = [_ z / x ]_ B ) ) /\ x = z ) -> w =/= v )
44	43	neneqd 2631	. . . . . . . . . . . . . . . . 17 |- ( ( ( w =/= v /\ ( w = B /\ v = [_ z / x ]_ B ) ) /\ x = z ) -> -. w = v )
45	42, 44	pm2.65da 580	. . . . . . . . . . . . . . . 16 |- ( ( w =/= v /\ ( w = B /\ v = [_ z / x ]_ B ) ) -> -. x = z )
46	45	neqned 2633	. . . . . . . . . . . . . . 15 |- ( ( w =/= v /\ ( w = B /\ v = [_ z / x ]_ B ) ) -> x =/= z )
47	46	adantlr 722	. . . . . . . . . . . . . 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 2459	. . . . . . . . . . . . . . . . . 18 |- ( w = B -> B = w )
50	49	ad2antrl 735	. . . . . . . . . . . . . . . . 17 |- ( ( ( w i^i v ) =/= (/) /\ ( w = B /\ v = [_ z / x ]_ B ) ) -> B = w )
51	39	ad2antll 736	. . . . . . . . . . . . . . . . 17 |- ( ( ( w i^i v ) =/= (/) /\ ( w = B /\ v = [_ z / x ]_ B ) ) -> [_ z / x ]_ B = v )
52	50, 51	ineq12d 3637	. . . . . . . . . . . . . . . 16 |- ( ( ( w i^i v ) =/= (/) /\ ( w = B /\ v = [_ z / x ]_ B ) ) -> ( B i^i [_ z / x ]_ B ) = ( w i^i v ) )
53		simpl 459	. . . . . . . . . . . . . . . 16 |- ( ( ( w i^i v ) =/= (/) /\ ( w = B /\ v = [_ z / x ]_ B ) ) -> ( w i^i v ) =/= (/) )
54	52, 53	eqnetrd 2693	. . . . . . . . . . . . . . 15 |- ( ( ( w i^i v ) =/= (/) /\ ( w = B /\ v = [_ z / x ]_ B ) ) -> ( B i^i [_ z / x ]_ B ) =/= (/) )
55	54	adantll 721	. . . . . . . . . . . . . 14 |- ( ( ( w =/= v /\ ( w i^i v ) =/= (/) ) /\ ( w = B /\ v = [_ z / x ]_ B ) ) -> ( B i^i [_ z / x ]_ B ) =/= (/) )
56	47, 55	jca 535	. . . . . . . . . . . . 13 |- ( ( ( w =/= v /\ ( w i^i v ) =/= (/) ) /\ ( w = B /\ v = [_ z / x ]_ B ) ) -> ( x =/= z /\ ( B i^i [_ z / x ]_ B ) =/= (/) ) )
57	56	ex 436	. . . . . . . . . . . 12 |- ( ( w =/= v /\ ( w i^i v ) =/= (/) ) -> ( ( w = B /\ v = [_ z / x ]_ B ) -> ( x =/= z /\ ( B i^i [_ z / x ]_ B ) =/= (/) ) ) )
58	57	adantl 468	. . . . . . . . . . 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 2863	. . . . . . . . . 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 26	. . . . . . . . 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 2858	. . . . . . . 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 436	. . . . . 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 2887	. . . 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 2594	. . . . . 6 |- F/_ u B
68		nfcsb1v 3381	. . . . . 6 |- F/_ x [_ u / x ]_ B
69		csbeq1a 3374	. . . . . 6 |- ( x = u -> B = [_ u / x ]_ B )
70	67, 68, 69	cbvdisj 4386	. . . . 5 |- (Disj x e. A B <-> Disj u e. A [_ u / x ]_ B )
71	70	notbii 298	. . . 4 |- ( -. Disj x e. A B <-> -. Disj u e. A [_ u / x ]_ B )
72		csbeq1a 3374	. . . . . . 7 |- ( u = z -> [_ u / x ]_ B = [_ z / u ]_ [_ u / x ]_ B )
73		csbco 3375	. . . . . . . 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 2487	. . . . . 6 |- ( u = z -> [_ u / x ]_ B = [_ z / x ]_ B )
76	75	ndisj2 37399	. . . . 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 2594	. . . . . . 7 |- F/_ x A
78		nfv 1763	. . . . . . . 8 |- F/ x u =/= z
79	68, 13	nfin 3641	. . . . . . . . 9 |- F/_ x ( [_ u / x ]_ B i^i [_ z / x ]_ B )
80		nfcv 2594	. . . . . . . . 9 |- F/_ x (/)
81	79, 80	nfne 2725	. . . . . . . 8 |- F/ x ( [_ u / x ]_ B i^i [_ z / x ]_ B ) =/= (/)
82	78, 81	nfan 2013	. . . . . . 7 |- F/ x ( u =/= z /\ ( [_ u / x ]_ B i^i [_ z / x ]_ B ) =/= (/) )
83	77, 82	nfrex 2852	. . . . . 6 |- F/ x E. z e. A ( u =/= z /\ ( [_ u / x ]_ B i^i [_ z / x ]_ B ) =/= (/) )
84		nfv 1763	. . . . . 6 |- F/ u E. z e. A ( x =/= z /\ ( B i^i [_ z / x ]_ B ) =/= (/) )
85		neeq1 2688	. . . . . . . 8 |- ( u = x -> ( u =/= z <-> x =/= z ) )
86		csbeq1 3368	. . . . . . . . . . 11 |- ( u = x -> [_ u / x ]_ B = [_ x / x ]_ B )
87		csbid 3373	. . . . . . . . . . . 12 |- [_ x / x ]_ B = B
88	87	a1i 11	. . . . . . . . . . 11 |- ( u = x -> [_ x / x ]_ B = B )
89	86, 88	eqtrd 2487	. . . . . . . . . 10 |- ( u = x -> [_ u / x ]_ B = B )
90	89	ineq1d 3635	. . . . . . . . 9 |- ( u = x -> ( [_ u / x ]_ B i^i [_ z / x ]_ B ) = ( B i^i [_ z / x ]_ B ) )
91	90	neeq1d 2685	. . . . . . . 8 |- ( u = x -> ( ( [_ u / x ]_ B i^i [_ z / x ]_ B ) =/= (/) <-> ( B i^i [_ z / x ]_ B ) =/= (/) ) )
92	85, 91	anbi12d 718	. . . . . . 7 |- ( u = x -> ( ( u =/= z /\ ( [_ u / x ]_ B i^i [_ z / x ]_ B ) =/= (/) ) <-> ( x =/= z /\ ( B i^i [_ z / x ]_ B ) =/= (/) ) ) )
93	92	rexbidv 2903	. . . . . 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 3018	. . . . 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 253	. . . 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 253	. . 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 216	. 2 |- ( -. Disj y e. ran F y -> -. Disj x e. A B )
98	97	con4i 134	1 |- (Disj x e. A B -> Disj y e. ran F y )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 371   = wceq 1446   e. wcel 1889   =/= wne 2624   E.wrex 2740   [_csb 3365   i^i cin 3405   (/)c0 3733  Disj wdisj 4376   |-> cmpt 4464   ran crn 4838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pr 4642

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-disj 4377  df-br 4406  df-opab 4465  df-mpt 4466  df-cnv 4845  df-dm 4847  df-rn 4848

This theorem is referenced by:  meadjiun  38314