Theorem ofrval 6141

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 2194	. . . . . 6 |- ( ( ph /\ x e. A ) -> ( F ` x ) = ( F ` x ) )
7		eqidd 2194	. . . . . 6 |- ( ( ph /\ x e. B ) -> ( G ` x ) = ( G ` x ) )
8	1, 2, 3, 4, 5, 6, 7	ofrfval 6139	. . . . 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 5554	. . . . . 6 |- ( x = X -> ( F ` x ) = ( F ` X ) )
11		fveq2 5554	. . . . . 6 |- ( x = X -> ( G ` x ) = ( G ` X ) )
12	10, 11	breq12d 4042	. . . . 5 |- ( x = X -> ( ( F ` x ) R ( G ` x ) <-> ( F ` X ) R ( G ` X ) ) )
13	12	rspccv 2861	. . . 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 1202	. 2 |- ( ( ph /\ F oR R G /\ X e. S ) -> ( F ` X ) R ( G ` X ) )
16		simp1 999	. . 3 |- ( ( ph /\ F oR R G /\ X e. S ) -> ph )
17		inss1 3379	. . . . 5 |- ( A i^i B ) C_ A
18	5, 17	eqsstrri 3212	. . . 4 |- S C_ A
19		simp3 1001	. . . 4 |- ( ( ph /\ F oR R G /\ X e. S ) -> X e. S )
20	18, 19	sselid 3177	. . 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 3380	. . . . 5 |- ( A i^i B ) C_ B
24	5, 23	eqsstrri 3212	. . . 4 |- S C_ B
25	24, 19	sselid 3177	. . 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 4060	1 |- ( ( ph /\ F oR R G /\ X e. S ) -> C R D )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2164   A.wral 2472   i^i cin 3152   class class class wbr 4029   Fn wfn 5249   ` cfv 5254   oRcofr 6129

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 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ofr 6131

This theorem is referenced by:  psrbaglesuppg  14158