Theorem fliftfund 5937

Description: The function F is the unique function defined by F ` A = B, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)

Hypotheses

Ref	Expression
flift.1	|- F = ran ( x e. X |-> <. A , B >. )
flift.2	|- ( ( ph /\ x e. X ) -> A e. R )
flift.3	|- ( ( ph /\ x e. X ) -> B e. S )
fliftfun.4	|- ( x = y -> A = C )
fliftfun.5	|- ( x = y -> B = D )
fliftfund.6	|- ( ( ph /\ ( x e. X /\ y e. X /\ A = C ) ) -> B = D )

Assertion

Ref	Expression
fliftfund	|- ( ph -> Fun F )

Distinct variable groups:   y, A   y, B   x, C   x, y, R   x, D   y, F   ph, x, y   x, X, y   x, S, y

Allowed substitution hints:   A( x)   B( x)   C( y)   D( y)   F( x)

Proof of Theorem fliftfund

Step	Hyp	Ref	Expression
1		fliftfund.6	. . . . 5 |- ( ( ph /\ ( x e. X /\ y e. X /\ A = C ) ) -> B = D )
2	1	3exp2 1251	. . . 4 |- ( ph -> ( x e. X -> ( y e. X -> ( A = C -> B = D ) ) ) )
3	2	imp32 257	. . 3 |- ( ( ph /\ ( x e. X /\ y e. X ) ) -> ( A = C -> B = D ) )
4	3	ralrimivva 2614	. 2 |- ( ph -> A. x e. X A. y e. X ( A = C -> B = D ) )
5		flift.1	. . 3 |- F = ran ( x e. X |-> <. A , B >. )
6		flift.2	. . 3 |- ( ( ph /\ x e. X ) -> A e. R )
7		flift.3	. . 3 |- ( ( ph /\ x e. X ) -> B e. S )
8		fliftfun.4	. . 3 |- ( x = y -> A = C )
9		fliftfun.5	. . 3 |- ( x = y -> B = D )
10	5, 6, 7, 8, 9	fliftfun 5936	. 2 |- ( ph -> ( Fun F <-> A. x e. X A. y e. X ( A = C -> B = D ) ) )
11	4, 10	mpbird 167	1 |- ( ph -> Fun F )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1004   = wceq 1397   e. wcel 2202   A.wral 2510   <.cop 3672   |-> cmpt 4150   ran crn 4726   Fun wfun 5320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334

This theorem is referenced by: (None)