Theorem funcnv4mpt 29470

Description: Two ways to say that a function in maps-to notation is single-rooted. (Contributed by Thierry Arnoux, 2-Mar-2017.)

Hypotheses

Ref	Expression
funcnvmpt.0	|- F/ x ph
funcnvmpt.1	|- F/_ x A
funcnvmpt.2	|- F/_ x F
funcnvmpt.3	|- F = ( x e. A |-> B )
funcnvmpt.4	|- ( ( ph /\ x e. A ) -> B e. V )

Assertion

Ref	Expression
funcnv4mpt	|- ( ph -> ( Fun `' F <-> A. i e. A A. j e. A ( i = j \/ [_ i / x ]_ B =/= [_ j / x ]_ B ) ) )

Distinct variable groups:   i, j, x   A, i, j   B, i, j   i, F   x, V   ph, i, j

Allowed substitution hints:   ph( x)   A( x)   B( x)   F( x, j)   V( i, j)

Proof of Theorem funcnv4mpt

Step	Hyp	Ref	Expression
1		nfv 1843	. 2 |- F/ i ph
2		nfcv 2764	. 2 |- F/_ i A
3		nfcv 2764	. 2 |- F/_ i F
4		funcnvmpt.3	. . 3 |- F = ( x e. A |-> B )
5		funcnvmpt.1	. . . 4 |- F/_ x A
6		nfcv 2764	. . . 4 |- F/_ i B
7		nfcsb1v 3549	. . . 4 |- F/_ x [_ i / x ]_ B
8		csbeq1a 3542	. . . 4 |- ( x = i -> B = [_ i / x ]_ B )
9	5, 2, 6, 7, 8	cbvmptf 4748	. . 3 |- ( x e. A |-> B ) = ( i e. A |-> [_ i / x ]_ B )
10	4, 9	eqtri 2644	. 2 |- F = ( i e. A |-> [_ i / x ]_ B )
11		funcnvmpt.4	. . . 4 |- ( ( ph /\ x e. A ) -> B e. V )
12	11	sbimi 1886	. . 3 |- ( [ i / x ] ( ph /\ x e. A ) -> [ i / x ] B e. V )
13		funcnvmpt.0	. . . . 5 |- F/ x ph
14		nfcv 2764	. . . . . 6 |- F/_ x i
15	14, 5	nfel 2777	. . . . 5 |- F/ x i e. A
16	13, 15	nfan 1828	. . . 4 |- F/ x ( ph /\ i e. A )
17		eleq1 2689	. . . . 5 |- ( x = i -> ( x e. A <-> i e. A ) )
18	17	anbi2d 740	. . . 4 |- ( x = i -> ( ( ph /\ x e. A ) <-> ( ph /\ i e. A ) ) )
19	16, 18	sbie 2408	. . 3 |- ( [ i / x ] ( ph /\ x e. A ) <-> ( ph /\ i e. A ) )
20		nfcv 2764	. . . . 5 |- F/_ x V
21	7, 20	nfel 2777	. . . 4 |- F/ x [_ i / x ]_ B e. V
22	8	eleq1d 2686	. . . 4 |- ( x = i -> ( B e. V <-> [_ i / x ]_ B e. V ) )
23	21, 22	sbie 2408	. . 3 |- ( [ i / x ] B e. V <-> [_ i / x ]_ B e. V )
24	12, 19, 23	3imtr3i 280	. 2 |- ( ( ph /\ i e. A ) -> [_ i / x ]_ B e. V )
25		csbeq1 3536	. 2 |- ( i = j -> [_ i / x ]_ B = [_ j / x ]_ B )
26	1, 2, 3, 10, 24, 25	funcnv5mpt 29469	1 |- ( ph -> ( Fun `' F <-> A. i e. A A. j e. A ( i = j \/ [_ i / x ]_ B =/= [_ j / x ]_ B ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   F/wnf 1708   [wsb 1880   e. wcel 1990   F/_wnfc 2751   =/= wne 2794   A.wral 2912   [_csb 3533   |-> cmpt 4729   `'ccnv 5113   Fun wfun 5882

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-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-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-fv 5896

This theorem is referenced by:  disjdsct  29480