Theorem fliftval 5817

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 276	. 2 |- ( ( ph /\ Y e. X ) -> Fun F )
3		simpr 110	. . . 4 |- ( ( ph /\ Y e. X ) -> Y e. X )
4		eqidd 2190	. . . . 5 |- ( ph -> D = D )
5		eqidd 2190	. . . . 5 |- ( Y e. X -> C = C )
6	4, 5	anim12ci 339	. . . 4 |- ( ( ph /\ Y e. X ) -> ( C = C /\ D = D ) )
7		fliftval.4	. . . . . . 7 |- ( x = Y -> A = C )
8	7	eqeq2d 2201	. . . . . 6 |- ( x = Y -> ( C = A <-> C = C ) )
9		fliftval.5	. . . . . . 7 |- ( x = Y -> B = D )
10	9	eqeq2d 2201	. . . . . 6 |- ( x = Y -> ( D = B <-> D = D ) )
11	8, 10	anbi12d 473	. . . . 5 |- ( x = Y -> ( ( C = A /\ D = B ) <-> ( C = C /\ D = D ) ) )
12	11	rspcev 2856	. . . 4 |- ( ( Y e. X /\ ( C = C /\ D = D ) ) -> E. x e. X ( C = A /\ D = B ) )
13	3, 6, 12	syl2anc 411	. . 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 5810	. . . 4 |- ( ph -> ( C F D <-> E. x e. X ( C = A /\ D = B ) ) )
18	17	adantr 276	. . 3 |- ( ( ph /\ Y e. X ) -> ( C F D <-> E. x e. X ( C = A /\ D = B ) ) )
19	13, 18	mpbird 167	. 2 |- ( ( ph /\ Y e. X ) -> C F D )
20		funbrfv 5570	. 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 104   <-> wb 105   = wceq 1364   e. wcel 2160   E.wrex 2469   <.cop 3610   class class class wbr 4018   |-> cmpt 4079   ran crn 4642   Fun wfun 5225   ` cfv 5231

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-iota 5193  df-fun 5233  df-fv 5239

This theorem is referenced by:  qliftval  6639