Theorem fliftfund 5800

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 1225	. . . 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 2559	. 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 5799	. 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 978   = wceq 1353   e. wcel 2148   A.wral 2455   <.cop 3597   |-> cmpt 4066   ran crn 4629   Fun wfun 5212

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226

This theorem is referenced by: (None)