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Theorem funfvop 3742
Description: Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41.
Assertion
Ref Expression
funfvop |- ((Fun F /\ A e. dom F) -> <.A, (F` A)>. e. F)

Proof of Theorem funfvop
StepHypRef Expression
1 fvex 3671 . . 3 |- (F` A) e. V
21isseti 1790 . 2 |- E.x x = (F` A)
3 visset 1788 . . . . . . 7 |- x e. V
43funopfvb 3695 . . . . . 6 |- ((Fun F /\ A e. dom F) -> ((F` A) = x <-> <.A, x>. e. F))
5 opeq2 2457 . . . . . . . 8 |- ((F` A) = x -> <.A, (F` A)>. = <.A, x>.)
65eleq1d 1516 . . . . . . 7 |- ((F` A) = x -> (<.A, (F` A)>. e. F <-> <.A, x>. e. F))
76biimprcd 156 . . . . . 6 |- (<.A, x>. e. F -> ((F` A) = x -> <.A, (F` A)>. e. F))
84, 7syl6bi 214 . . . . 5 |- ((Fun F /\ A e. dom F) -> ((F` A) = x -> ((F` A) = x -> <.A, (F` A)>. e. F)))
98pm2.43d 65 . . . 4 |- ((Fun F /\ A e. dom F) -> ((F` A) = x -> <.A, (F` A)>. e. F))
10 eqcom 1453 . . . 4 |- (x = (F` A) <-> (F` A) = x)
119, 10syl5ib 206 . . 3 |- ((Fun F /\ A e. dom F) -> (x = (F` A) -> <.A, (F` A)>. e. F))
121119.23adv 1198 . 2 |- ((Fun F /\ A e. dom F) -> (E.x x = (F` A) -> <.A, (F` A)>. e. F))
132, 12mpi 44 1 |- ((Fun F /\ A e. dom F) -> <.A, (F` A)>. e. F)
Colors of variables: wff set class
Syntax hints:   -> wi 3   /\ wa 223  E.wex 956   = wceq 1099   e. wcel 1105  <.cop 2382  dom cdm 3133  Fun wfun 3139  ` cfv 3145
This theorem is referenced by:  fvimacnv 3744  fnopfv 3750  fvelrn 3751  dff2 3756  funfvima3 3793  fundmen 4363  adjt 9987
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-13 1107  ax-14 1108  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436  ax-sep 2671  ax-nul 2678  ax-pow 2710  ax-pr 2747  ax-un 2830
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360  df-clab 1441  df-cleq 1446  df-clel 1449  df-ne 1563  df-rex 1626  df-v 1787  df-dif 2020  df-un 2021  df-in 2022  df-ss 2024  df-nul 2252  df-pw 2373  df-sn 2383  df-pr 2384  df-op 2387  df-uni 2472  df-br 2588  df-opab 2635  df-id 2797  df-xp 3147  df-cnv 3149  df-co 3150  df-dm 3151  df-rn 3152  df-res 3153  df-ima 3154  df-fun 3155  df-fn 3156  df-fv 3161
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