Theorem eqfnoprval 4011

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 3792	. 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 3719	. . . . . 6 |- (z = <.x, y>. -> (F` z) = (F` <.x, y>.))
3		fveq2 3719	. . . . . 6 |- (z = <.x, y>. -> (G` z) = (G` <.x, y>.))
4	2, 3	eqeq12d 1487	. . . . 5 |- (z = <.x, y>. -> ((F` z) = (G` z) <-> (F` <.x, y>.) = (G` <.x, y>.)))
5		df-opr 3960	. . . . . 6 |- (xFy) = (F` <.x, y>.)
6		df-opr 3960	. . . . . 6 |- (xGy) = (G` <.x, y>.)
7	5, 6	eqeq12i 1486	. . . . 5 |- ((xFy) = (xGy) <-> (F` <.x, y>.) = (G` <.x, y>.))
8	4, 7	syl6bbr 537	. . . 4 |- (z = <.x, y>. -> ((F` z) = (G` z) <-> (xFy) = (xGy)))
9	8	ralxp 3214	. . 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 535	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 955  A.wral 1643  <.cop 2408   X. cxp 3164   Fn wfn 3173  ` cfv 3178  (class class class)co 3958

This theorem is referenced by:  dfseq0 6508  sspg 8349  ssps 8351  sspmlem 8353  hhip 8999

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-fv 3194  df-opr 3960

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