Theorem fliftval 5709

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 2141	. . . . 5 |- ( ph -> D = D )
5		eqidd 2141	. . . . 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 2152	. . . . . 6 |- ( x = Y -> ( C = A <-> C = C ) )
9		fliftval.5	. . . . . . 7 |- ( x = Y -> B = D )
10	9	eqeq2d 2152	. . . . . 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 2793	. . . 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 5702	. . . 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 5468	. 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 1332   e. wcel 1481   E.wrex 2418   <.cop 3535   class class class wbr 3937   |-> cmpt 3997   ran crn 4548   Fun wfun 5125   ` cfv 5131

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-iota 5096  df-fun 5133  df-fv 5139

This theorem is referenced by:  qliftval  6523