Theorem eqfnoprval 4016

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

Assertion

Ref	Expression
eqfnoprval	|- ((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 eqfnoprval

Step	Hyp	Ref	Expression
1		eqfnfv 3797	. 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 3724	. . . . . 6 |- (z = <.x, y>. -> (F` z) = (F` <.x, y>.))
3		fveq2 3724	. . . . . 6 |- (z = <.x, y>. -> (G` z) = (G` <.x, y>.))
4	2, 3	eqeq12d 1489	. . . . 5 |- (z = <.x, y>. -> ((F` z) = (G` z) <-> (F` <.x, y>.) = (G` <.x, y>.)))
5		df-opr 3965	. . . . . 6 |- (xFy) = (F` <.x, y>.)
6		df-opr 3965	. . . . . 6 |- (xGy) = (G` <.x, y>.)
7	5, 6	eqeq12i 1488	. . . . 5 |- ((xFy) = (xGy) <-> (F` <.x, y>.) = (G` <.x, y>.))
8	4, 7	syl6bbr 538	. . . 4 |- (z = <.x, y>. -> ((F` z) = (G` z) <-> (xFy) = (xGy)))
9	8	ralxp 3218	. . 3 |- (A.z e. (A X. B)(F` z) = (G` z) <-> A.x e. A A.y e. B (xFy) = (xGy))
10	9	anbi2i 480	. 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 536	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 146   /\ wa 223   = wceq 956  A.wral 1645  <.cop 2411   X. cxp 3168   Fn wfn 3177  ` cfv 3182  (class class class)co 3963

This theorem is referenced by:  dfseq0 6563  sspg 8387  ssps 8389  sspmlem 8391  hhip 9044

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198  df-opr 3965

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