Theorem fliftfund 5948

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 1252	. . . 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 2615	. 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 5947	. 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 1005   = wceq 1398   e. wcel 2202   A.wral 2511   <.cop 3676   |-> cmpt 4155   ran crn 4732   Fun wfun 5327

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341

This theorem is referenced by: (None)