Theorem qliftval 6731

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 6723	. 2 |- ( ( ph /\ x e. X ) -> [ x ] R e. ( X /. R ) )
6		eceq1 6678	. 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 5892	1 |- ( ( ph /\ C e. X ) -> ( F ` [ C ] R ) = B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2178   _Vcvv 2776   <.cop 3646   |-> cmpt 4121   ran crn 4694   Fun wfun 5284   ` cfv 5290   Er wer 6640   [cec 6641   /.cqs 6642

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fv 5298  df-er 6643  df-ec 6645  df-qs 6649

This theorem is referenced by: (None)