Theorem funopfvg 2854

Description: The second element in an ordered pair member of a function is the function's value.

Assertion

Ref	Expression
funopfvg	|- ((B e. C /\ Fun F) -> (<.A, B>. e. F -> (F` A) = B))

Proof of Theorem funopfvg

Step	Hyp	Ref	Expression
1		opeq2 1877	. . . . . 6 |- (x = B -> <.A, x>. = <.A, B>.)
2	1	eleq1d 1155	. . . . 5 |- (x = B -> (<.A, x>. e. F <-> <.A, B>. e. F))
3		cleq2 1110	. . . . 5 |- (x = B -> ((F` A) = x <-> (F` A) = B))
4	2, 3	imbi12d 474	. . . 4 |- (x = B -> ((<.A, x>. e. F -> (F` A) = x) <-> (<.A, B>. e. F -> (F` A) = B)))
5	4	imbi2d 464	. . 3 |- (x = B -> ((Fun F -> (<.A, x>. e. F -> (F` A) = x)) <-> (Fun F -> (<.A, B>. e. F -> (F` A) = B))))
6		visset 1350	. . . 4 |- x e. V
7	6	funfvopi 2853	. . 3 |- (Fun F -> (<.A, x>. e. F -> (F` A) = x))
8	5, 7	vtoclg 1383	. 2 |- (B e. C -> (Fun F -> (<.A, B>. e. F -> (F` A) = B)))
9	8	imp 277	1 |- ((B e. C /\ Fun F) -> (<.A, B>. e. F -> (F` A) = B))

Syntax hints:   -> wi 2   /\ wa 196   = wceq 1091   e. wcel 1092  <.cop 1810  Fun wfun 2416  ` cfv 2422

This theorem is referenced by:  fvopab3ig 2869  oprabvalig 3048

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fv 2438

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