Theorem fliftfund 5921

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 1249	. . . 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 2612	. 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 5920	. 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 1002   = wceq 1395   e. wcel 2200   A.wral 2508   <.cop 3669   |-> cmpt 4145   ran crn 4720   Fun wfun 5312

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326

This theorem is referenced by: (None)