Theorem eqfnoprv 4945

Description: Equality of two operations is determined by their values.

Assertion

Ref	Expression
eqfnoprv	|- ((F Fn (A X. B) /\ G Fn (C X. D)) -> (F = G <-> ((A X. B) = (C X. D) /\ A.x e. A A.y e. B (xFy) = (xGy))))

Distinct variable groups:   x,y,A   x,B,y   x,F,y   x,G,y

Proof of Theorem eqfnoprv

Step	Hyp	Ref	Expression
1		eqfnfv 4766	. 2 |- ((F Fn (A X. B) /\ G Fn (C X. D)) -> (F = G <-> ((A X. B) = (C X. D) /\ A.z e. (A X. B)(F` z) = (G` z))))
2		fveq2 4681	. . . . . 6 |- (z = <.x, y>. -> (F` z) = (F` <.x, y>.))
3		fveq2 4681	. . . . . 6 |- (z = <.x, y>. -> (G` z) = (G` <.x, y>.))
4	2, 3	eqeq12d 1899	. . . . 5 |- (z = <.x, y>. -> ((F` z) = (G` z) <-> (F` <.x, y>.) = (G` <.x, y>.)))
5		df-opr 4886	. . . . . 6 |- (xFy) = (F` <.x, y>.)
6		df-opr 4886	. . . . . 6 |- (xGy) = (G` <.x, y>.)
7	5, 6	eqeq12i 1897	. . . . 5 |- ((xFy) = (xGy) <-> (F` <.x, y>.) = (G` <.x, y>.))
8	4, 7	syl6bbr 597	. . . 4 |- (z = <.x, y>. -> ((F` z) = (G` z) <-> (xFy) = (xGy)))
9	8	ralxp 4041	. . 3 |- (A.z e. (A X. B)(F` z) = (G` z) <-> A.x e. A A.y e. B (xFy) = (xGy))
10	9	anbi2i 538	. 2 |- (((A X. B) = (C X. D) /\ A.z e. (A X. B)(F` z) = (G` z)) <-> ((A X. B) = (C X. D) /\ A.x e. A A.y e. B (xFy) = (xGy)))
11	1, 10	syl6bb 595	1 |- ((F Fn (A X. B) /\ G Fn (C X. D)) -> (F = G <-> ((A X. B) = (C X. D) /\ A.x e. A A.y e. B (xFy) = (xGy))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298  A.wral 2105  <.cop 3046   X. cxp 3984   Fn wfn 3993  ` cfv 3998  (class class class)co 4884

This theorem is referenced by:  dfseq0 7806  sspg 9726  ssps 9728  sspmlem 9730  hhip 10677  eqfnoprv2 15704

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-opr 4886

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