Theorem funcnvmptOLD 29467

Description: Condition for a function in maps-to notation to be single-rooted. (Contributed by Thierry Arnoux, 28-Feb-2017.) (New usage is discouraged.) (Proof modification is discouraged.)

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
funcnvmptOLD	|- ( ph -> ( Fun `' F <-> A. y E* x ( 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 funcnvmptOLD

Step	Hyp	Ref	Expression
1		relcnv 5503	. . . 4 |- Rel `' F
2		nfcv 2764	. . . . 5 |- F/_ y `' F
3		funcnvmpt.2	. . . . . 6 |- F/_ x F
4	3	nfcnv 5301	. . . . 5 |- F/_ x `' F
5	2, 4	dffun6f 5902	. . . 4 |- ( Fun `' F <-> ( Rel `' F /\ A. y E* x y `' F x ) )
6	1, 5	mpbiran 953	. . 3 |- ( Fun `' F <-> A. y E* x y `' F x )
7		vex 3203	. . . . . 6 |- y e. _V
8		vex 3203	. . . . . 6 |- x e. _V
9	7, 8	brcnv 5305	. . . . 5 |- ( y `' F x <-> x F y )
10	9	mobii 2493	. . . 4 |- ( E* x y `' F x <-> E* x x F y )
11	10	albii 1747	. . 3 |- ( A. y E* x y `' F x <-> A. y E* x x F y )
12	6, 11	bitri 264	. 2 |- ( Fun `' F <-> A. y E* x x F y )
13		nfv 1843	. . 3 |- F/ y ph
14		funcnvmpt.0	. . . 4 |- F/ x ph
15		funmpt 5926	. . . . . . . . 9 |- Fun ( x e. A |-> B )
16		funcnvmpt.3	. . . . . . . . . 10 |- F = ( x e. A |-> B )
17	16	funeqi 5909	. . . . . . . . 9 |- ( Fun F <-> Fun ( x e. A |-> B ) )
18	15, 17	mpbir 221	. . . . . . . 8 |- Fun F
19		funbrfv2b 6240	. . . . . . . 8 |- ( Fun F -> ( x F y <-> ( x e. dom F /\ ( F ` x ) = y ) ) )
20	18, 19	ax-mp 5	. . . . . . 7 |- ( x F y <-> ( x e. dom F /\ ( F ` x ) = y ) )
21		funcnvmpt.4	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. A ) -> B e. V )
22		elex 3212	. . . . . . . . . . . . . 14 |- ( B e. V -> B e. _V )
23	21, 22	syl 17	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. A ) -> B e. _V )
24	23	ex 450	. . . . . . . . . . . 12 |- ( ph -> ( x e. A -> B e. _V ) )
25	14, 24	ralrimi 2957	. . . . . . . . . . 11 |- ( ph -> A. x e. A B e. _V )
26		funcnvmpt.1	. . . . . . . . . . . 12 |- F/_ x A
27	26	rabid2f 3119	. . . . . . . . . . 11 |- ( A = { x e. A | B e. _V } <-> A. x e. A B e. _V )
28	25, 27	sylibr 224	. . . . . . . . . 10 |- ( ph -> A = { x e. A | B e. _V } )
29	16	dmmpt 5630	. . . . . . . . . 10 |- dom F = { x e. A | B e. _V }
30	28, 29	syl6reqr 2675	. . . . . . . . 9 |- ( ph -> dom F = A )
31	30	eleq2d 2687	. . . . . . . 8 |- ( ph -> ( x e. dom F <-> x e. A ) )
32	31	anbi1d 741	. . . . . . 7 |- ( ph -> ( ( x e. dom F /\ ( F ` x ) = y ) <-> ( x e. A /\ ( F ` x ) = y ) ) )
33	20, 32	syl5bb 272	. . . . . 6 |- ( ph -> ( x F y <-> ( x e. A /\ ( F ` x ) = y ) ) )
34	33	bian1d 29306	. . . . 5 |- ( ph -> ( ( x e. A /\ x F y ) <-> ( x e. A /\ ( F ` x ) = y ) ) )
35	16	fveq1i 6192	. . . . . . . . . 10 |- ( F ` x ) = ( ( x e. A |-> B ) ` x )
36		simpr 477	. . . . . . . . . . 11 |- ( ( ph /\ x e. A ) -> x e. A )
37	26	fvmpt2f 6283	. . . . . . . . . . 11 |- ( ( x e. A /\ B e. V ) -> ( ( x e. A |-> B ) ` x ) = B )
38	36, 21, 37	syl2anc 693	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> ( ( x e. A |-> B ) ` x ) = B )
39	35, 38	syl5eq 2668	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> ( F ` x ) = B )
40	39	eqeq2d 2632	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> ( y = ( F ` x ) <-> y = B ) )
41	31	biimpar 502	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> x e. dom F )
42		funbrfvb 6238	. . . . . . . . . 10 |- ( ( Fun F /\ x e. dom F ) -> ( ( F ` x ) = y <-> x F y ) )
43	18, 41, 42	sylancr 695	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> ( ( F ` x ) = y <-> x F y ) )
44		eqcom 2629	. . . . . . . . . . 11 |- ( ( F ` x ) = y <-> y = ( F ` x ) )
45	44	bibi1i 328	. . . . . . . . . 10 |- ( ( ( F ` x ) = y <-> x F y ) <-> ( y = ( F ` x ) <-> x F y ) )
46	45	imbi2i 326	. . . . . . . . 9 |- ( ( ( ph /\ x e. A ) -> ( ( F ` x ) = y <-> x F y ) ) <-> ( ( ph /\ x e. A ) -> ( y = ( F ` x ) <-> x F y ) ) )
47	43, 46	mpbi 220	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> ( y = ( F ` x ) <-> x F y ) )
48	40, 47	bitr3d 270	. . . . . . 7 |- ( ( ph /\ x e. A ) -> ( y = B <-> x F y ) )
49	48	ex 450	. . . . . 6 |- ( ph -> ( x e. A -> ( y = B <-> x F y ) ) )
50	49	pm5.32d 671	. . . . 5 |- ( ph -> ( ( x e. A /\ y = B ) <-> ( x e. A /\ x F y ) ) )
51	34, 50, 33	3bitr4rd 301	. . . 4 |- ( ph -> ( x F y <-> ( x e. A /\ y = B ) ) )
52	14, 51	mobid 2489	. . 3 |- ( ph -> ( E* x x F y <-> E* x ( x e. A /\ y = B ) ) )
53	13, 52	albid 2090	. 2 |- ( ph -> ( A. y E* x x F y <-> A. y E* x ( x e. A /\ y = B ) ) )
54	12, 53	syl5bb 272	1 |- ( ph -> ( Fun `' F <-> A. y E* x ( x e. A /\ y = B ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   F/wnf 1708   e. wcel 1990   E*wmo 2471   F/_wnfc 2751   A.wral 2912   {crab 2916   _Vcvv 3200   class class class wbr 4653   |-> cmpt 4729   `'ccnv 5113   dom cdm 5114   Rel wrel 5119   Fun wfun 5882   ` cfv 5888

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-ral 2917  df-rex 2918  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: (None)