Theorem fnoprval 4017

Description: Representation of an operation class abstraction in terms of its values.

Assertion

Ref	Expression
fnoprval	|- (F Fn (A X. B) <-> F = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = (xFy))})

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

Proof of Theorem fnoprval

Step	Hyp	Ref	Expression
1		fnopabfv 3758	. 2 |- (F Fn (A X. B) <-> F = {<.w, z>. | (w e. (A X. B) /\ z = (F` w))})
2		elxp 3202	. . . . . . 7 |- (w e. (A X. B) <-> E.xE.y(w = <.x, y>. /\ (x e. A /\ y e. B)))
3	2	anbi1i 481	. . . . . 6 |- ((w e. (A X. B) /\ z = (F` w)) <-> (E.xE.y(w = <.x, y>. /\ (x e. A /\ y e. B)) /\ z = (F` w)))
4		19.41vv 1306	. . . . . 6 |- (E.xE.y((w = <.x, y>. /\ (x e. A /\ y e. B)) /\ z = (F` w)) <-> (E.xE.y(w = <.x, y>. /\ (x e. A /\ y e. B)) /\ z = (F` w)))
5		anass 439	. . . . . . . 8 |- (((w = <.x, y>. /\ (x e. A /\ y e. B)) /\ z = (F` w)) <-> (w = <.x, y>. /\ ((x e. A /\ y e. B) /\ z = (F` w))))
6		fveq2 3724	. . . . . . . . . . . 12 |- (w = <.x, y>. -> (F` w) = (F` <.x, y>.))
7		df-opr 3965	. . . . . . . . . . . 12 |- (xFy) = (F` <.x, y>.)
8	6, 7	syl6eqr 1525	. . . . . . . . . . 11 |- (w = <.x, y>. -> (F` w) = (xFy))
9	8	eqeq2d 1486	. . . . . . . . . 10 |- (w = <.x, y>. -> (z = (F` w) <-> z = (xFy)))
10	9	anbi2d 616	. . . . . . . . 9 |- (w = <.x, y>. -> (((x e. A /\ y e. B) /\ z = (F` w)) <-> ((x e. A /\ y e. B) /\ z = (xFy))))
11	10	pm5.32i 645	. . . . . . . 8 |- ((w = <.x, y>. /\ ((x e. A /\ y e. B) /\ z = (F` w))) <-> (w = <.x, y>. /\ ((x e. A /\ y e. B) /\ z = (xFy))))
12	5, 11	bitr 173	. . . . . . 7 |- (((w = <.x, y>. /\ (x e. A /\ y e. B)) /\ z = (F` w)) <-> (w = <.x, y>. /\ ((x e. A /\ y e. B) /\ z = (xFy))))
13	12	2exbii 1052	. . . . . 6 |- (E.xE.y((w = <.x, y>. /\ (x e. A /\ y e. B)) /\ z = (F` w)) <-> E.xE.y(w = <.x, y>. /\ ((x e. A /\ y e. B) /\ z = (xFy))))
14	3, 4, 13	3bitr2 179	. . . . 5 |- ((w e. (A X. B) /\ z = (F` w)) <-> E.xE.y(w = <.x, y>. /\ ((x e. A /\ y e. B) /\ z = (xFy))))
15	14	opabbii 2671	. . . 4 |- {<.w, z>. | (w e. (A X. B) /\ z = (F` w))} = {<.w, z>. | E.xE.y(w = <.x, y>. /\ ((x e. A /\ y e. B) /\ z = (xFy)))}
16		dfoprab2 3991	. . . 4 |- {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = (xFy))} = {<.w, z>. | E.xE.y(w = <.x, y>. /\ ((x e. A /\ y e. B) /\ z = (xFy)))}
17	15, 16	eqtr4 1498	. . 3 |- {<.w, z>. | (w e. (A X. B) /\ z = (F` w))} = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = (xFy))}
18	17	eqeq2i 1485	. 2 |- (F = {<.w, z>. | (w e. (A X. B) /\ z = (F` w))} <-> F = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = (xFy))})
19	1, 18	bitr 173	1 |- (F Fn (A X. B) <-> F = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = (xFy))})

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  <.cop 2411  {copab 2666   X. cxp 3168   Fn wfn 3177  ` cfv 3182  (class class class)co 3963  {copab2 3964

This theorem is referenced by:  foprval 4018  mapxpen 4495  dfioo2 6403  cnnvm 8313

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-9 965  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  ax-un 2866

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  df-oprab 3966

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