Theorem ofrval 5985

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
ofrval.6	|- ( ( ph /\ X e. A ) -> ( F ` X ) = C )
ofrval.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 2138	. . . . . 6 |- ( ( ph /\ x e. A ) -> ( F ` x ) = ( F ` x ) )
7		eqidd 2138	. . . . . 6 |- ( ( ph /\ x e. B ) -> ( G ` x ) = ( G ` x ) )
8	1, 2, 3, 4, 5, 6, 7	ofrfval 5983	. . . . 5 |- ( ph -> ( F oR R G <-> A. x e. S ( F ` x ) R ( G ` x ) ) )
9	8	biimpa 294	. . . 4 |- ( ( ph /\ F oR R G ) -> A. x e. S ( F ` x ) R ( G ` x ) )
10		fveq2 5414	. . . . . 6 |- ( x = X -> ( F ` x ) = ( F ` X ) )
11		fveq2 5414	. . . . . 6 |- ( x = X -> ( G ` x ) = ( G ` X ) )
12	10, 11	breq12d 3937	. . . . 5 |- ( x = X -> ( ( F ` x ) R ( G ` x ) <-> ( F ` X ) R ( G ` X ) ) )
13	12	rspccv 2781	. . . 4 |- ( A. x e. S ( F ` x ) R ( G ` x ) -> ( X e. S -> ( F ` X ) R ( G ` X ) ) )
14	9, 13	syl 14	. . 3 |- ( ( ph /\ F oR R G ) -> ( X e. S -> ( F ` X ) R ( G ` X ) ) )
15	14	3impia 1178	. 2 |- ( ( ph /\ F oR R G /\ X e. S ) -> ( F ` X ) R ( G ` X ) )
16		simp1 981	. . 3 |- ( ( ph /\ F oR R G /\ X e. S ) -> ph )
17		inss1 3291	. . . . 5 |- ( A i^i B ) C_ A
18	5, 17	eqsstrri 3125	. . . 4 |- S C_ A
19		simp3 983	. . . 4 |- ( ( ph /\ F oR R G /\ X e. S ) -> X e. S )
20	18, 19	sseldi 3090	. . 3 |- ( ( ph /\ F oR R G /\ X e. S ) -> X e. A )
21		ofrval.6	. . 3 |- ( ( ph /\ X e. A ) -> ( F ` X ) = C )
22	16, 20, 21	syl2anc 408	. 2 |- ( ( ph /\ F oR R G /\ X e. S ) -> ( F ` X ) = C )
23		inss2 3292	. . . . 5 |- ( A i^i B ) C_ B
24	5, 23	eqsstrri 3125	. . . 4 |- S C_ B
25	24, 19	sseldi 3090	. . 3 |- ( ( ph /\ F oR R G /\ X e. S ) -> X e. B )
26		ofrval.7	. . 3 |- ( ( ph /\ X e. B ) -> ( G ` X ) = D )
27	16, 25, 26	syl2anc 408	. 2 |- ( ( ph /\ F oR R G /\ X e. S ) -> ( G ` X ) = D )
28	15, 22, 27	3brtr3d 3954	1 |- ( ( ph /\ F oR R G /\ X e. S ) -> C R D )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 962   = wceq 1331   e. wcel 1480   A.wral 2414   i^i cin 3065   class class class wbr 3924   Fn wfn 5113   ` cfv 5118   oRcofr 5974

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ofr 5976

This theorem is referenced by: (None)