Theorem fliftfund 6563

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 1285	. . . 4 |- ( ph -> ( x e. X -> ( y e. X -> ( A = C -> B = D ) ) ) )
3	2	imp32 449	. . 3 |- ( ( ph /\ ( x e. X /\ y e. X ) ) -> ( A = C -> B = D ) )
4	3	ralrimivva 2971	. 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 6562	. 2 |- ( ph -> ( Fun F <-> A. x e. X A. y e. X ( A = C -> B = D ) ) )
11	4, 10	mpbird 247	1 |- ( ph -> Fun F )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   <.cop 4183   |-> cmpt 4729   ran crn 5115   Fun wfun 5882

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-ne 2795  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-f 5892  df-fv 5896

This theorem is referenced by:  cygznlem2a  19916  pi1xfrf  22853  pi1cof  22859