Theorem ofrval 6245

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 2232	. . . . . 6 |- ( ( ph /\ x e. A ) -> ( F ` x ) = ( F ` x ) )
7		eqidd 2232	. . . . . 6 |- ( ( ph /\ x e. B ) -> ( G ` x ) = ( G ` x ) )
8	1, 2, 3, 4, 5, 6, 7	ofrfval 6243	. . . . 5 |- ( ph -> ( F oR R G <-> A. x e. S ( F ` x ) R ( G ` x ) ) )
9	8	biimpa 296	. . . 4 |- ( ( ph /\ F oR R G ) -> A. x e. S ( F ` x ) R ( G ` x ) )
10		fveq2 5639	. . . . . 6 |- ( x = X -> ( F ` x ) = ( F ` X ) )
11		fveq2 5639	. . . . . 6 |- ( x = X -> ( G ` x ) = ( G ` X ) )
12	10, 11	breq12d 4101	. . . . 5 |- ( x = X -> ( ( F ` x ) R ( G ` x ) <-> ( F ` X ) R ( G ` X ) ) )
13	12	rspccv 2907	. . . 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 1226	. 2 |- ( ( ph /\ F oR R G /\ X e. S ) -> ( F ` X ) R ( G ` X ) )
16		simp1 1023	. . 3 |- ( ( ph /\ F oR R G /\ X e. S ) -> ph )
17		inss1 3427	. . . . 5 |- ( A i^i B ) C_ A
18	5, 17	eqsstrri 3260	. . . 4 |- S C_ A
19		simp3 1025	. . . 4 |- ( ( ph /\ F oR R G /\ X e. S ) -> X e. S )
20	18, 19	sselid 3225	. . 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 411	. 2 |- ( ( ph /\ F oR R G /\ X e. S ) -> ( F ` X ) = C )
23		inss2 3428	. . . . 5 |- ( A i^i B ) C_ B
24	5, 23	eqsstrri 3260	. . . 4 |- S C_ B
25	24, 19	sselid 3225	. . 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 411	. 2 |- ( ( ph /\ F oR R G /\ X e. S ) -> ( G ` X ) = D )
28	15, 22, 27	3brtr3d 4119	1 |- ( ( ph /\ F oR R G /\ X e. S ) -> C R D )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1004   = wceq 1397   e. wcel 2202   A.wral 2510   i^i cin 3199   class class class wbr 4088   Fn wfn 5321   ` cfv 5326   oRcofr 6233

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ofr 6235

This theorem is referenced by:  psrbaglesuppg  14685