Theorem qliftval 6251

Description: The value of the function F. (Contributed by Mario Carneiro, 23-Dec-2016.)

Hypotheses

Ref	Expression
qlift.1	|- F = ran ( x e. X |-> <. [ x ] R , A >. )
qlift.2	|- ( ( ph /\ x e. X ) -> A e. Y )
qlift.3	|- ( ph -> R Er X )
qlift.4	|- ( ph -> X e. _V )
qliftval.4	|- ( x = C -> A = B )
qliftval.6	|- ( ph -> Fun F )

Assertion

Ref	Expression
qliftval	|- ( ( ph /\ C e. X ) -> ( F ` [ C ] R ) = B )

Distinct variable groups:   x, B   x, C   ph, x   x, R   x, X   x, Y

Allowed substitution hints:   A( x)   F( x)

Proof of Theorem qliftval

Step	Hyp	Ref	Expression
1		qlift.1	. 2 |- F = ran ( x e. X |-> <. [ x ] R , A >. )
2		qlift.2	. . 3 |- ( ( ph /\ x e. X ) -> A e. Y )
3		qlift.3	. . 3 |- ( ph -> R Er X )
4		qlift.4	. . 3 |- ( ph -> X e. _V )
5	1, 2, 3, 4	qliftlem 6243	. 2 |- ( ( ph /\ x e. X ) -> [ x ] R e. ( X /. R ) )
6		eceq1 6200	. 2 |- ( x = C -> [ x ] R = [ C ] R )
7		qliftval.4	. 2 |- ( x = C -> A = B )
8		qliftval.6	. 2 |- ( ph -> Fun F )
9	1, 5, 2, 6, 7, 8	fliftval 5465	1 |- ( ( ph /\ C e. X ) -> ( F ` [ C ] R ) = B )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1285   e. wcel 1434   _Vcvv 2602   <.cop 3403   |-> cmpt 3841   ran crn 4366   Fun wfun 4920   ` cfv 4926   Er wer 6162   [cec 6163   /.cqs 6164

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966  ax-un 4190

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-br 3788  df-opab 3842  df-mpt 3843  df-id 4050  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378  df-iota 4891  df-fun 4928  df-fv 4934  df-er 6165  df-ec 6167  df-qs 6171

This theorem is referenced by: (None)