Theorem funcnvmpt 28264

Description: Condition for a function in maps-to notation to be single-rooted. (Contributed by Thierry Arnoux, 28-Feb-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
funcnvmpt	|- ( ph -> ( Fun `' F <-> A. y E* x e. A y = B ) )

Distinct variable groups:   x, y   y, F   ph, y

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

Proof of Theorem funcnvmpt

Step	Hyp	Ref	Expression
1		relcnv 5206	. . . 4 |- Rel `' F
2		nfcv 2591	. . . . 5 |- F/_ y `' F
3		funcnvmpt.2	. . . . . 6 |- F/_ x F
4	3	nfcnv 5012	. . . . 5 |- F/_ x `' F
5	2, 4	dffun6f 5595	. . . 4 |- ( Fun `' F <-> ( Rel `' F /\ A. y E* x y `' F x ) )
6	1, 5	mpbiran 928	. . 3 |- ( Fun `' F <-> A. y E* x y `' F x )
7		vex 3047	. . . . . 6 |- y e. _V
8		vex 3047	. . . . . 6 |- x e. _V
9	7, 8	brcnv 5016	. . . . 5 |- ( y `' F x <-> x F y )
10	9	mobii 2321	. . . 4 |- ( E* x y `' F x <-> E* x x F y )
11	10	albii 1690	. . 3 |- ( A. y E* x y `' F x <-> A. y E* x x F y )
12	6, 11	bitri 253	. 2 |- ( Fun `' F <-> A. y E* x x F y )
13		funcnvmpt.0	. . . . 5 |- F/ x ph
14		funcnvmpt.3	. . . . . . . . . 10 |- F = ( x e. A |-> B )
15	14	funmpt2 5618	. . . . . . . . 9 |- Fun F
16		funbrfv2b 5907	. . . . . . . . 9 |- ( Fun F -> ( x F y <-> ( x e. dom F /\ ( F ` x ) = y ) ) )
17	15, 16	ax-mp 5	. . . . . . . 8 |- ( x F y <-> ( x e. dom F /\ ( F ` x ) = y ) )
18		funcnvmpt.4	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. A ) -> B e. V )
19		elex 3053	. . . . . . . . . . . . . . 15 |- ( B e. V -> B e. _V )
20	18, 19	syl 17	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. A ) -> B e. _V )
21	20	ex 436	. . . . . . . . . . . . 13 |- ( ph -> ( x e. A -> B e. _V ) )
22	13, 21	ralrimi 2787	. . . . . . . . . . . 12 |- ( ph -> A. x e. A B e. _V )
23		funcnvmpt.1	. . . . . . . . . . . . 13 |- F/_ x A
24	23	rabid2f 28129	. . . . . . . . . . . 12 |- ( A = { x e. A | B e. _V } <-> A. x e. A B e. _V )
25	22, 24	sylibr 216	. . . . . . . . . . 11 |- ( ph -> A = { x e. A | B e. _V } )
26	14	dmmpt 5329	. . . . . . . . . . 11 |- dom F = { x e. A | B e. _V }
27	25, 26	syl6reqr 2503	. . . . . . . . . 10 |- ( ph -> dom F = A )
28	27	eleq2d 2513	. . . . . . . . 9 |- ( ph -> ( x e. dom F <-> x e. A ) )
29	28	anbi1d 710	. . . . . . . 8 |- ( ph -> ( ( x e. dom F /\ ( F ` x ) = y ) <-> ( x e. A /\ ( F ` x ) = y ) ) )
30	17, 29	syl5bb 261	. . . . . . 7 |- ( ph -> ( x F y <-> ( x e. A /\ ( F ` x ) = y ) ) )
31	30	bian1d 28093	. . . . . 6 |- ( ph -> ( ( x e. A /\ x F y ) <-> ( x e. A /\ ( F ` x ) = y ) ) )
32		simpr 463	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> x e. A )
33	14	fveq1i 5864	. . . . . . . . . . 11 |- ( F ` x ) = ( ( x e. A |-> B ) ` x )
34	23	fvmpt2f 5947	. . . . . . . . . . 11 |- ( ( x e. A /\ B e. V ) -> ( ( x e. A |-> B ) ` x ) = B )
35	33, 34	syl5eq 2496	. . . . . . . . . 10 |- ( ( x e. A /\ B e. V ) -> ( F ` x ) = B )
36	32, 18, 35	syl2anc 666	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> ( F ` x ) = B )
37	36	eqeq2d 2460	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> ( y = ( F ` x ) <-> y = B ) )
38		eqcom 2457	. . . . . . . . 9 |- ( ( F ` x ) = y <-> y = ( F ` x ) )
39	28	biimpar 488	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> x e. dom F )
40		funbrfvb 5905	. . . . . . . . . 10 |- ( ( Fun F /\ x e. dom F ) -> ( ( F ` x ) = y <-> x F y ) )
41	15, 39, 40	sylancr 668	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> ( ( F ` x ) = y <-> x F y ) )
42	38, 41	syl5bbr 263	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> ( y = ( F ` x ) <-> x F y ) )
43	37, 42	bitr3d 259	. . . . . . 7 |- ( ( ph /\ x e. A ) -> ( y = B <-> x F y ) )
44	43	pm5.32da 646	. . . . . 6 |- ( ph -> ( ( x e. A /\ y = B ) <-> ( x e. A /\ x F y ) ) )
45	31, 44, 30	3bitr4rd 290	. . . . 5 |- ( ph -> ( x F y <-> ( x e. A /\ y = B ) ) )
46	13, 45	mobid 2317	. . . 4 |- ( ph -> ( E* x x F y <-> E* x ( x e. A /\ y = B ) ) )
47		df-rmo 2744	. . . 4 |- ( E* x e. A y = B <-> E* x ( x e. A /\ y = B ) )
48	46, 47	syl6bbr 267	. . 3 |- ( ph -> ( E* x x F y <-> E* x e. A y = B ) )
49	48	albidv 1766	. 2 |- ( ph -> ( A. y E* x x F y <-> A. y E* x e. A y = B ) )
50	12, 49	syl5bb 261	1 |- ( ph -> ( Fun `' F <-> A. y E* x e. A y = B ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   A.wal 1441   = wceq 1443   F/wnf 1666   e. wcel 1886   E*wmo 2299   F/_wnfc 2578   A.wral 2736   E*wrmo 2739   {crab 2740   _Vcvv 3044   class class class wbr 4401   |-> cmpt 4460   `'ccnv 4832   dom cdm 4833   Rel wrel 4838   Fun wfun 5575   ` cfv 5581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pr 4638

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-fv 5589

This theorem is referenced by:  funcnv5mpt  28265