Theorem funcnvmpt 27210

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 5374	. . . 4 |- Rel `' F
2		nfcv 2629	. . . . 5 |- F/_ y `' F
3		funcnvmpt.2	. . . . . 6 |- F/_ x F
4	3	nfcnv 5181	. . . . 5 |- F/_ x `' F
5	2, 4	dffun6f 5602	. . . 4 |- ( Fun `' F <-> ( Rel `' F /\ A. y E* x y `' F x ) )
6	1, 5	mpbiran 916	. . 3 |- ( Fun `' F <-> A. y E* x y `' F x )
7		vex 3116	. . . . . 6 |- y e. _V
8		vex 3116	. . . . . 6 |- x e. _V
9	7, 8	brcnv 5185	. . . . 5 |- ( y `' F x <-> x F y )
10	9	mobii 2301	. . . 4 |- ( E* x y `' F x <-> E* x x F y )
11	10	albii 1620	. . 3 |- ( A. y E* x y `' F x <-> A. y E* x x F y )
12	6, 11	bitri 249	. 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 5625	. . . . . . . . 9 |- Fun F
16		funbrfv2b 5912	. . . . . . . . 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 3122	. . . . . . . . . . . . . . 15 |- ( B e. V -> B e. _V )
20	18, 19	syl 16	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. A ) -> B e. _V )
21	20	ex 434	. . . . . . . . . . . . 13 |- ( ph -> ( x e. A -> B e. _V ) )
22	13, 21	ralrimi 2864	. . . . . . . . . . . 12 |- ( ph -> A. x e. A B e. _V )
23		funcnvmpt.1	. . . . . . . . . . . . 13 |- F/_ x A
24	23	rabid2f 27104	. . . . . . . . . . . 12 |- ( A = { x e. A | B e. _V } <-> A. x e. A B e. _V )
25	22, 24	sylibr 212	. . . . . . . . . . 11 |- ( ph -> A = { x e. A | B e. _V } )
26	14	dmmpt 5502	. . . . . . . . . . 11 |- dom F = { x e. A | B e. _V }
27	25, 26	syl6reqr 2527	. . . . . . . . . 10 |- ( ph -> dom F = A )
28	27	eleq2d 2537	. . . . . . . . 9 |- ( ph -> ( x e. dom F <-> x e. A ) )
29	28	anbi1d 704	. . . . . . . 8 |- ( ph -> ( ( x e. dom F /\ ( F ` x ) = y ) <-> ( x e. A /\ ( F ` x ) = y ) ) )
30	17, 29	syl5bb 257	. . . . . . 7 |- ( ph -> ( x F y <-> ( x e. A /\ ( F ` x ) = y ) ) )
31	30	bian1d 27070	. . . . . 6 |- ( ph -> ( ( x e. A /\ x F y ) <-> ( x e. A /\ ( F ` x ) = y ) ) )
32		simpr 461	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> x e. A )
33	14	fveq1i 5867	. . . . . . . . . . 11 |- ( F ` x ) = ( ( x e. A |-> B ) ` x )
34	23	fvmpt2f 27198	. . . . . . . . . . 11 |- ( ( x e. A /\ B e. V ) -> ( ( x e. A |-> B ) ` x ) = B )
35	33, 34	syl5eq 2520	. . . . . . . . . 10 |- ( ( x e. A /\ B e. V ) -> ( F ` x ) = B )
36	32, 18, 35	syl2anc 661	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> ( F ` x ) = B )
37	36	eqeq2d 2481	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> ( y = ( F ` x ) <-> y = B ) )
38		eqcom 2476	. . . . . . . . 9 |- ( ( F ` x ) = y <-> y = ( F ` x ) )
39	28	biimpar 485	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> x e. dom F )
40		funbrfvb 5910	. . . . . . . . . 10 |- ( ( Fun F /\ x e. dom F ) -> ( ( F ` x ) = y <-> x F y ) )
41	15, 39, 40	sylancr 663	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> ( ( F ` x ) = y <-> x F y ) )
42	38, 41	syl5bbr 259	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> ( y = ( F ` x ) <-> x F y ) )
43	37, 42	bitr3d 255	. . . . . . 7 |- ( ( ph /\ x e. A ) -> ( y = B <-> x F y ) )
44	43	pm5.32da 641	. . . . . 6 |- ( ph -> ( ( x e. A /\ y = B ) <-> ( x e. A /\ x F y ) ) )
45	31, 44, 30	3bitr4rd 286	. . . . 5 |- ( ph -> ( x F y <-> ( x e. A /\ y = B ) ) )
46	13, 45	mobid 2297	. . . 4 |- ( ph -> ( E* x x F y <-> E* x ( x e. A /\ y = B ) ) )
47		df-rmo 2822	. . . 4 |- ( E* x e. A y = B <-> E* x ( x e. A /\ y = B ) )
48	46, 47	syl6bbr 263	. . 3 |- ( ph -> ( E* x x F y <-> E* x e. A y = B ) )
49	48	albidv 1689	. 2 |- ( ph -> ( A. y E* x x F y <-> A. y E* x e. A y = B ) )
50	12, 49	syl5bb 257	1 |- ( ph -> ( Fun `' F <-> A. y E* x e. A y = B ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   A.wal 1377   = wceq 1379   F/wnf 1599   e. wcel 1767   E*wmo 2276   F/_wnfc 2615   A.wral 2814   E*wrmo 2817   {crab 2818   _Vcvv 3113   class class class wbr 4447   |-> cmpt 4505   `'ccnv 4998   dom cdm 4999   Rel wrel 5004   Fun wfun 5582   ` cfv 5588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-fv 5596

This theorem is referenced by:  funcnv5mpt  27211