Theorem fliftval 5768

Description: The value of the function F. (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 )
fliftval.4	|- ( x = Y -> A = C )
fliftval.5	|- ( x = Y -> B = D )
fliftval.6	|- ( ph -> Fun F )

Assertion

Ref	Expression
fliftval	|- ( ( ph /\ Y e. X ) -> ( F ` C ) = D )

Distinct variable groups:   x, C   x, R   x, Y   x, D   ph, x   x, X   x, S

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

Proof of Theorem fliftval

Step	Hyp	Ref	Expression
1		fliftval.6	. . 3 |- ( ph -> Fun F )
2	1	adantr 274	. 2 |- ( ( ph /\ Y e. X ) -> Fun F )
3		simpr 109	. . . 4 |- ( ( ph /\ Y e. X ) -> Y e. X )
4		eqidd 2166	. . . . 5 |- ( ph -> D = D )
5		eqidd 2166	. . . . 5 |- ( Y e. X -> C = C )
6	4, 5	anim12ci 337	. . . 4 |- ( ( ph /\ Y e. X ) -> ( C = C /\ D = D ) )
7		fliftval.4	. . . . . . 7 |- ( x = Y -> A = C )
8	7	eqeq2d 2177	. . . . . 6 |- ( x = Y -> ( C = A <-> C = C ) )
9		fliftval.5	. . . . . . 7 |- ( x = Y -> B = D )
10	9	eqeq2d 2177	. . . . . 6 |- ( x = Y -> ( D = B <-> D = D ) )
11	8, 10	anbi12d 465	. . . . 5 |- ( x = Y -> ( ( C = A /\ D = B ) <-> ( C = C /\ D = D ) ) )
12	11	rspcev 2830	. . . 4 |- ( ( Y e. X /\ ( C = C /\ D = D ) ) -> E. x e. X ( C = A /\ D = B ) )
13	3, 6, 12	syl2anc 409	. . 3 |- ( ( ph /\ Y e. X ) -> E. x e. X ( C = A /\ D = B ) )
14		flift.1	. . . . 5 |- F = ran ( x e. X |-> <. A , B >. )
15		flift.2	. . . . 5 |- ( ( ph /\ x e. X ) -> A e. R )
16		flift.3	. . . . 5 |- ( ( ph /\ x e. X ) -> B e. S )
17	14, 15, 16	fliftel 5761	. . . 4 |- ( ph -> ( C F D <-> E. x e. X ( C = A /\ D = B ) ) )
18	17	adantr 274	. . 3 |- ( ( ph /\ Y e. X ) -> ( C F D <-> E. x e. X ( C = A /\ D = B ) ) )
19	13, 18	mpbird 166	. 2 |- ( ( ph /\ Y e. X ) -> C F D )
20		funbrfv 5525	. 2 |- ( Fun F -> ( C F D -> ( F ` C ) = D ) )
21	2, 19, 20	sylc 62	1 |- ( ( ph /\ Y e. X ) -> ( F ` C ) = D )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   E.wrex 2445   <.cop 3579   class class class wbr 3982   |-> cmpt 4043   ran crn 4605   Fun wfun 5182   ` cfv 5188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-iota 5153  df-fun 5190  df-fv 5196

This theorem is referenced by:  qliftval  6587