Theorem ofrval 6573

Description: Exhibit a function relation at a point. (Contributed by Mario Carneiro, 28-Jul-2014.)

Hypotheses

Ref	Expression
offval.1	|- ( ph -> F Fn A )
offval.2	|- ( ph -> G Fn B )
offval.3	|- ( ph -> A e. V )
offval.4	|- ( ph -> B e. W )
offval.5	|- ( A i^i B ) = S
ofval.6	|- ( ( ph /\ X e. A ) -> ( F ` X ) = C )
ofval.7	|- ( ( ph /\ X e. B ) -> ( G ` X ) = D )

Assertion

Ref	Expression
ofrval	|- ( ( ph /\ F oR R G /\ X e. S ) -> C R D )

Proof of Theorem ofrval

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		offval.1	. . . . . 6 |- ( ph -> F Fn A )
2		offval.2	. . . . . 6 |- ( ph -> G Fn B )
3		offval.3	. . . . . 6 |- ( ph -> A e. V )
4		offval.4	. . . . . 6 |- ( ph -> B e. W )
5		offval.5	. . . . . 6 |- ( A i^i B ) = S
6		eqidd 2463	. . . . . 6 |- ( ( ph /\ x e. A ) -> ( F ` x ) = ( F ` x ) )
7		eqidd 2463	. . . . . 6 |- ( ( ph /\ x e. B ) -> ( G ` x ) = ( G ` x ) )
8	1, 2, 3, 4, 5, 6, 7	ofrfval 6571	. . . . 5 |- ( ph -> ( F oR R G <-> A. x e. S ( F ` x ) R ( G ` x ) ) )
9	8	biimpa 491	. . . 4 |- ( ( ph /\ F oR R G ) -> A. x e. S ( F ` x ) R ( G ` x ) )
10		fveq2 5892	. . . . . 6 |- ( x = X -> ( F ` x ) = ( F ` X ) )
11		fveq2 5892	. . . . . 6 |- ( x = X -> ( G ` x ) = ( G ` X ) )
12	10, 11	breq12d 4431	. . . . 5 |- ( x = X -> ( ( F ` x ) R ( G ` x ) <-> ( F ` X ) R ( G ` X ) ) )
13	12	rspccv 3159	. . . 4 |- ( A. x e. S ( F ` x ) R ( G ` x ) -> ( X e. S -> ( F ` X ) R ( G ` X ) ) )
14	9, 13	syl 17	. . 3 |- ( ( ph /\ F oR R G ) -> ( X e. S -> ( F ` X ) R ( G ` X ) ) )
15	14	3impia 1212	. 2 |- ( ( ph /\ F oR R G /\ X e. S ) -> ( F ` X ) R ( G ` X ) )
16		simp1 1014	. . 3 |- ( ( ph /\ F oR R G /\ X e. S ) -> ph )
17		inss1 3664	. . . . 5 |- ( A i^i B ) C_ A
18	5, 17	eqsstr3i 3475	. . . 4 |- S C_ A
19		simp3 1016	. . . 4 |- ( ( ph /\ F oR R G /\ X e. S ) -> X e. S )
20	18, 19	sseldi 3442	. . 3 |- ( ( ph /\ F oR R G /\ X e. S ) -> X e. A )
21		ofval.6	. . 3 |- ( ( ph /\ X e. A ) -> ( F ` X ) = C )
22	16, 20, 21	syl2anc 671	. 2 |- ( ( ph /\ F oR R G /\ X e. S ) -> ( F ` X ) = C )
23		inss2 3665	. . . . 5 |- ( A i^i B ) C_ B
24	5, 23	eqsstr3i 3475	. . . 4 |- S C_ B
25	24, 19	sseldi 3442	. . 3 |- ( ( ph /\ F oR R G /\ X e. S ) -> X e. B )
26		ofval.7	. . 3 |- ( ( ph /\ X e. B ) -> ( G ` X ) = D )
27	16, 25, 26	syl2anc 671	. 2 |- ( ( ph /\ F oR R G /\ X e. S ) -> ( G ` X ) = D )
28	15, 22, 27	3brtr3d 4448	1 |- ( ( ph /\ F oR R G /\ X e. S ) -> C R D )

Syntax hints:   -> wi 4   /\ wa 375   /\ w3a 991   = wceq 1455   e. wcel 1898   A.wral 2749   i^i cin 3415   class class class wbr 4418   Fn wfn 5600   ` cfv 5605   oRcofr 6562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4531  ax-sep 4541  ax-nul 4550  ax-pr 4656

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-iun 4294  df-br 4419  df-opab 4478  df-mpt 4479  df-id 4771  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-ofr 6564

This theorem is referenced by:  itg1le  22727  gsumle  28593  ftc1anclem5  32067