Theorem fnopabfv 3753

Description: Representation of a function in terms of its values.

Assertion

Ref	Expression
fnopabfv	|- (F Fn A <-> F = {<.x, y>. | (x e. A /\ y = (F` x))})

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

Proof of Theorem fnopabfv

Step	Hyp	Ref	Expression
1		fnop 3587	. . . . . . . 8 |- ((F Fn A /\ <.z, w>. e. F) -> z e. A)
2	1	ex 373	. . . . . . 7 |- (F Fn A -> (<.z, w>. e. F -> z e. A))
3	2	pm4.71rd 638	. . . . . 6 |- (F Fn A -> (<.z, w>. e. F <-> (z e. A /\ <.z, w>. e. F)))
4		visset 1810	. . . . . . . . 9 |- w e. V
5	4	fnopfvb 3749	. . . . . . . 8 |- ((F Fn A /\ z e. A) -> ((F` z) = w <-> <.z, w>. e. F))
6		eqcom 1475	. . . . . . . 8 |- (w = (F` z) <-> (F` z) = w)
7	5, 6	syl5bb 531	. . . . . . 7 |- ((F Fn A /\ z e. A) -> (w = (F` z) <-> <.z, w>. e. F))
8	7	pm5.32da 648	. . . . . 6 |- (F Fn A -> ((z e. A /\ w = (F` z)) <-> (z e. A /\ <.z, w>. e. F)))
9	3, 8	bitr4d 530	. . . . 5 |- (F Fn A -> (<.z, w>. e. F <-> (z e. A /\ w = (F` z))))
10		visset 1810	. . . . . 6 |- z e. V
11		eleq1 1532	. . . . . . 7 |- (x = z -> (x e. A <-> z e. A))
12		fveq2 3719	. . . . . . . 8 |- (x = z -> (F` x) = (F` z))
13	12	eqeq2d 1484	. . . . . . 7 |- (x = z -> (y = (F` x) <-> y = (F` z)))
14	11, 13	anbi12d 627	. . . . . 6 |- (x = z -> ((x e. A /\ y = (F` x)) <-> (z e. A /\ y = (F` z))))
15		eqeq1 1479	. . . . . . 7 |- (y = w -> (y = (F` z) <-> w = (F` z)))
16	15	anbi2d 615	. . . . . 6 |- (y = w -> ((z e. A /\ y = (F` z)) <-> (z e. A /\ w = (F` z))))
17	10, 4, 14, 16	opelopab 2816	. . . . 5 |- (<.z, w>. e. {<.x, y>. | (x e. A /\ y = (F` x))} <-> (z e. A /\ w = (F` z)))
18	9, 17	syl6bbr 537	. . . 4 |- (F Fn A -> (<.z, w>. e. F <-> <.z, w>. e. {<.x, y>. | (x e. A /\ y = (F` x))}))
19	18	19.21aivv 1286	. . 3 |- (F Fn A -> A.zA.w(<.z, w>. e. F <-> <.z, w>. e. {<.x, y>. | (x e. A /\ y = (F` x))}))
20		fnrel 3582	. . . . 5 |- (F Fn A -> Rel F)
21		relopab 3262	. . . . 5 |- Rel {<.x, y>. | (x e. A /\ y = (F` x))}
22	20, 21	jctir 293	. . . 4 |- (F Fn A -> (Rel F /\ Rel {<.x, y>. | (x e. A /\ y = (F` x))}))
23		eqrel 3246	. . . 4 |- ((Rel F /\ Rel {<.x, y>. | (x e. A /\ y = (F` x))}) -> (F = {<.x, y>. | (x e. A /\ y = (F` x))} <-> A.zA.w(<.z, w>. e. F <-> <.z, w>. e. {<.x, y>. | (x e. A /\ y = (F` x))})))
24	22, 23	syl 10	. . 3 |- (F Fn A -> (F = {<.x, y>. | (x e. A /\ y = (F` x))} <-> A.zA.w(<.z, w>. e. F <-> <.z, w>. e. {<.x, y>. | (x e. A /\ y = (F` x))})))
25	19, 24	mpbird 196	. 2 |- (F Fn A -> F = {<.x, y>. | (x e. A /\ y = (F` x))})
26		fvex 3727	. . . 4 |- (F` x) e. V
27		eqid 1474	. . . 4 |- {<.x, y>. | (x e. A /\ y = (F` x))} = {<.x, y>. | (x e. A /\ y = (F` x))}
28	26, 27	fnopab2 3614	. . 3 |- {<.x, y>. | (x e. A /\ y = (F` x))} Fn A
29		fneq1 3578	. . 3 |- (F = {<.x, y>. | (x e. A /\ y = (F` x))} -> (F Fn A <-> {<.x, y>. | (x e. A /\ y = (F` x))} Fn A))
30	28, 29	mpbiri 194	. 2 |- (F = {<.x, y>. | (x e. A /\ y = (F` x))} -> F Fn A)
31	25, 30	impbi 157	1 |- (F Fn A <-> F = {<.x, y>. | (x e. A /\ y = (F` x))})

Syntax hints:   <-> wb 146   /\ wa 223  A.wal 953   = wceq 955   e. wcel 957  <.cop 2408  {copab 2662  Rel wrel 3171   Fn wfn 3173  ` cfv 3178

This theorem is referenced by:  fopabfv 3826  fnoprval 4012  xpmapenlem3 4487  serzfsum 6957  xplm 7937  ip1cnilem2 8336  hilnorm 8985  pjrn 9604

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-9 964  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  ax-un 2862

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

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