Theorem funcnv5mpt 29469

Description: Two ways to say that a function in maps-to notation is single-rooted. (Contributed by Thierry Arnoux, 1-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 )
funcnv5mpt.1	|- ( x = z -> B = C )

Assertion

Ref	Expression
funcnv5mpt	|- ( ph -> ( Fun `' F <-> A. x e. A A. z e. A ( x = z \/ B =/= C ) ) )

Distinct variable groups:   x, z   ph, z   z, A   z, B   x, C

Allowed substitution hints:   ph( x)   A( x)   B( x)   C( z)   F( x, z)   V( x, z)

Proof of Theorem funcnv5mpt

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funcnvmpt.0	. . 3 |- F/ x ph
2		funcnvmpt.1	. . 3 |- F/_ x A
3		funcnvmpt.2	. . 3 |- F/_ x F
4		funcnvmpt.3	. . 3 |- F = ( x e. A |-> B )
5		funcnvmpt.4	. . 3 |- ( ( ph /\ x e. A ) -> B e. V )
6	1, 2, 3, 4, 5	funcnvmpt 29468	. 2 |- ( ph -> ( Fun `' F <-> A. y E* x e. A y = B ) )
7		nne 2798	. . . . . . . . 9 |- ( -. B =/= C <-> B = C )
8		eqvincg 3329	. . . . . . . . . 10 |- ( B e. V -> ( B = C <-> E. y ( y = B /\ y = C ) ) )
9	5, 8	syl 17	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> ( B = C <-> E. y ( y = B /\ y = C ) ) )
10	7, 9	syl5bb 272	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> ( -. B =/= C <-> E. y ( y = B /\ y = C ) ) )
11	10	imbi1d 331	. . . . . . 7 |- ( ( ph /\ x e. A ) -> ( ( -. B =/= C -> x = z ) <-> ( E. y ( y = B /\ y = C ) -> x = z ) ) )
12		orcom 402	. . . . . . . 8 |- ( ( x = z \/ B =/= C ) <-> ( B =/= C \/ x = z ) )
13		df-or 385	. . . . . . . 8 |- ( ( B =/= C \/ x = z ) <-> ( -. B =/= C -> x = z ) )
14	12, 13	bitri 264	. . . . . . 7 |- ( ( x = z \/ B =/= C ) <-> ( -. B =/= C -> x = z ) )
15		19.23v 1902	. . . . . . 7 |- ( A. y ( ( y = B /\ y = C ) -> x = z ) <-> ( E. y ( y = B /\ y = C ) -> x = z ) )
16	11, 14, 15	3bitr4g 303	. . . . . 6 |- ( ( ph /\ x e. A ) -> ( ( x = z \/ B =/= C ) <-> A. y ( ( y = B /\ y = C ) -> x = z ) ) )
17	16	ralbidv 2986	. . . . 5 |- ( ( ph /\ x e. A ) -> ( A. z e. A ( x = z \/ B =/= C ) <-> A. z e. A A. y ( ( y = B /\ y = C ) -> x = z ) ) )
18		ralcom4 3224	. . . . 5 |- ( A. z e. A A. y ( ( y = B /\ y = C ) -> x = z ) <-> A. y A. z e. A ( ( y = B /\ y = C ) -> x = z ) )
19	17, 18	syl6bb 276	. . . 4 |- ( ( ph /\ x e. A ) -> ( A. z e. A ( x = z \/ B =/= C ) <-> A. y A. z e. A ( ( y = B /\ y = C ) -> x = z ) ) )
20	1, 19	ralbida 2982	. . 3 |- ( ph -> ( A. x e. A A. z e. A ( x = z \/ B =/= C ) <-> A. x e. A A. y A. z e. A ( ( y = B /\ y = C ) -> x = z ) ) )
21		nfcv 2764	. . . . . 6 |- F/_ z A
22		nfv 1843	. . . . . 6 |- F/ x y = C
23		funcnv5mpt.1	. . . . . . 7 |- ( x = z -> B = C )
24	23	eqeq2d 2632	. . . . . 6 |- ( x = z -> ( y = B <-> y = C ) )
25	2, 21, 22, 24	rmo4f 29337	. . . . 5 |- ( E* x e. A y = B <-> A. x e. A A. z e. A ( ( y = B /\ y = C ) -> x = z ) )
26	25	albii 1747	. . . 4 |- ( A. y E* x e. A y = B <-> A. y A. x e. A A. z e. A ( ( y = B /\ y = C ) -> x = z ) )
27		ralcom4 3224	. . . 4 |- ( A. x e. A A. y A. z e. A ( ( y = B /\ y = C ) -> x = z ) <-> A. y A. x e. A A. z e. A ( ( y = B /\ y = C ) -> x = z ) )
28	26, 27	bitr4i 267	. . 3 |- ( A. y E* x e. A y = B <-> A. x e. A A. y A. z e. A ( ( y = B /\ y = C ) -> x = z ) )
29	20, 28	syl6bbr 278	. 2 |- ( ph -> ( A. x e. A A. z e. A ( x = z \/ B =/= C ) <-> A. y E* x e. A y = B ) )
30	6, 29	bitr4d 271	1 |- ( ph -> ( Fun `' F <-> A. x e. A A. z e. A ( x = z \/ B =/= C ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   F/wnf 1708   e. wcel 1990   F/_wnfc 2751   =/= wne 2794   A.wral 2912   E*wrmo 2915   |-> 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:  funcnv4mpt  29470