Theorem ofrval 6549

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 2458	. . . . . 6 |- ( ( ph /\ x e. A ) -> ( F ` x ) = ( F ` x ) )
7		eqidd 2458	. . . . . 6 |- ( ( ph /\ x e. B ) -> ( G ` x ) = ( G ` x ) )
8	1, 2, 3, 4, 5, 6, 7	ofrfval 6547	. . . . 5 |- ( ph -> ( F oR R G <-> A. x e. S ( F ` x ) R ( G ` x ) ) )
9	8	biimpa 484	. . . 4 |- ( ( ph /\ F oR R G ) -> A. x e. S ( F ` x ) R ( G ` x ) )
10		fveq2 5872	. . . . . 6 |- ( x = X -> ( F ` x ) = ( F ` X ) )
11		fveq2 5872	. . . . . 6 |- ( x = X -> ( G ` x ) = ( G ` X ) )
12	10, 11	breq12d 4469	. . . . 5 |- ( x = X -> ( ( F ` x ) R ( G ` x ) <-> ( F ` X ) R ( G ` X ) ) )
13	12	rspccv 3207	. . . 4 |- ( A. x e. S ( F ` x ) R ( G ` x ) -> ( X e. S -> ( F ` X ) R ( G ` X ) ) )
14	9, 13	syl 16	. . 3 |- ( ( ph /\ F oR R G ) -> ( X e. S -> ( F ` X ) R ( G ` X ) ) )
15	14	3impia 1193	. 2 |- ( ( ph /\ F oR R G /\ X e. S ) -> ( F ` X ) R ( G ` X ) )
16		simp1 996	. . 3 |- ( ( ph /\ F oR R G /\ X e. S ) -> ph )
17		inss1 3714	. . . . 5 |- ( A i^i B ) C_ A
18	5, 17	eqsstr3i 3530	. . . 4 |- S C_ A
19		simp3 998	. . . 4 |- ( ( ph /\ F oR R G /\ X e. S ) -> X e. S )
20	18, 19	sseldi 3497	. . 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 661	. 2 |- ( ( ph /\ F oR R G /\ X e. S ) -> ( F ` X ) = C )
23		inss2 3715	. . . . 5 |- ( A i^i B ) C_ B
24	5, 23	eqsstr3i 3530	. . . 4 |- S C_ B
25	24, 19	sseldi 3497	. . 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 661	. 2 |- ( ( ph /\ F oR R G /\ X e. S ) -> ( G ` X ) = D )
28	15, 22, 27	3brtr3d 4485	1 |- ( ( ph /\ F oR R G /\ X e. S ) -> C R D )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 973   = wceq 1395   e. wcel 1819   A.wral 2807   i^i cin 3470   class class class wbr 4456   Fn wfn 5589   ` cfv 5594   oRcofr 6538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pr 4695

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ofr 6540

This theorem is referenced by:  itg1le  22245  gsumle  27922  ftc1anclem5  30256