Theorem funcnvmpt 28137

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 5218	. . . 4 |- Rel `' F
2		nfcv 2582	. . . . 5 |- F/_ y `' F
3		funcnvmpt.2	. . . . . 6 |- F/_ x F
4	3	nfcnv 5024	. . . . 5 |- F/_ x `' F
5	2, 4	dffun6f 5606	. . . 4 |- ( Fun `' F <-> ( Rel `' F /\ A. y E* x y `' F x ) )
6	1, 5	mpbiran 926	. . 3 |- ( Fun `' F <-> A. y E* x y `' F x )
7		vex 3081	. . . . . 6 |- y e. _V
8		vex 3081	. . . . . 6 |- x e. _V
9	7, 8	brcnv 5028	. . . . 5 |- ( y `' F x <-> x F y )
10	9	mobii 2287	. . . 4 |- ( E* x y `' F x <-> E* x x F y )
11	10	albii 1687	. . 3 |- ( A. y E* x y `' F x <-> A. y E* x x F y )
12	6, 11	bitri 252	. 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 5629	. . . . . . . . 9 |- Fun F
16		funbrfv2b 5916	. . . . . . . . 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 3087	. . . . . . . . . . . . . . 15 |- ( B e. V -> B e. _V )
20	18, 19	syl 17	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. A ) -> B e. _V )
21	20	ex 435	. . . . . . . . . . . . 13 |- ( ph -> ( x e. A -> B e. _V ) )
22	13, 21	ralrimi 2823	. . . . . . . . . . . 12 |- ( ph -> A. x e. A B e. _V )
23		funcnvmpt.1	. . . . . . . . . . . . 13 |- F/_ x A
24	23	rabid2f 27998	. . . . . . . . . . . 12 |- ( A = { x e. A | B e. _V } <-> A. x e. A B e. _V )
25	22, 24	sylibr 215	. . . . . . . . . . 11 |- ( ph -> A = { x e. A | B e. _V } )
26	14	dmmpt 5341	. . . . . . . . . . 11 |- dom F = { x e. A | B e. _V }
27	25, 26	syl6reqr 2480	. . . . . . . . . 10 |- ( ph -> dom F = A )
28	27	eleq2d 2490	. . . . . . . . 9 |- ( ph -> ( x e. dom F <-> x e. A ) )
29	28	anbi1d 709	. . . . . . . 8 |- ( ph -> ( ( x e. dom F /\ ( F ` x ) = y ) <-> ( x e. A /\ ( F ` x ) = y ) ) )
30	17, 29	syl5bb 260	. . . . . . 7 |- ( ph -> ( x F y <-> ( x e. A /\ ( F ` x ) = y ) ) )
31	30	bian1d 27961	. . . . . 6 |- ( ph -> ( ( x e. A /\ x F y ) <-> ( x e. A /\ ( F ` x ) = y ) ) )
32		simpr 462	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> x e. A )
33	14	fveq1i 5873	. . . . . . . . . . 11 |- ( F ` x ) = ( ( x e. A |-> B ) ` x )
34	23	fvmpt2f 5956	. . . . . . . . . . 11 |- ( ( x e. A /\ B e. V ) -> ( ( x e. A |-> B ) ` x ) = B )
35	33, 34	syl5eq 2473	. . . . . . . . . 10 |- ( ( x e. A /\ B e. V ) -> ( F ` x ) = B )
36	32, 18, 35	syl2anc 665	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> ( F ` x ) = B )
37	36	eqeq2d 2434	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> ( y = ( F ` x ) <-> y = B ) )
38		eqcom 2429	. . . . . . . . 9 |- ( ( F ` x ) = y <-> y = ( F ` x ) )
39	28	biimpar 487	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> x e. dom F )
40		funbrfvb 5914	. . . . . . . . . 10 |- ( ( Fun F /\ x e. dom F ) -> ( ( F ` x ) = y <-> x F y ) )
41	15, 39, 40	sylancr 667	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> ( ( F ` x ) = y <-> x F y ) )
42	38, 41	syl5bbr 262	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> ( y = ( F ` x ) <-> x F y ) )
43	37, 42	bitr3d 258	. . . . . . 7 |- ( ( ph /\ x e. A ) -> ( y = B <-> x F y ) )
44	43	pm5.32da 645	. . . . . 6 |- ( ph -> ( ( x e. A /\ y = B ) <-> ( x e. A /\ x F y ) ) )
45	31, 44, 30	3bitr4rd 289	. . . . 5 |- ( ph -> ( x F y <-> ( x e. A /\ y = B ) ) )
46	13, 45	mobid 2283	. . . 4 |- ( ph -> ( E* x x F y <-> E* x ( x e. A /\ y = B ) ) )
47		df-rmo 2781	. . . 4 |- ( E* x e. A y = B <-> E* x ( x e. A /\ y = B ) )
48	46, 47	syl6bbr 266	. . 3 |- ( ph -> ( E* x x F y <-> E* x e. A y = B ) )
49	48	albidv 1757	. 2 |- ( ph -> ( A. y E* x x F y <-> A. y E* x e. A y = B ) )
50	12, 49	syl5bb 260	1 |- ( ph -> ( Fun `' F <-> A. y E* x e. A y = B ) )

Syntax hints:   -> wi 4   <-> wb 187   /\ wa 370   A.wal 1435   = wceq 1437   F/wnf 1663   e. wcel 1867   E*wmo 2264   F/_wnfc 2568   A.wral 2773   E*wrmo 2776   {crab 2777   _Vcvv 3078   class class class wbr 4417   |-> cmpt 4475   `'ccnv 4844   dom cdm 4845   Rel wrel 4850   Fun wfun 5586   ` cfv 5592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-fv 5600

This theorem is referenced by:  funcnv5mpt  28138