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Theorem mpofvexi 6206
Description: Sufficient condition for an operation maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)
Hypotheses
Ref Expression
fmpo.1 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
fnmpoi.2 𝐶 ∈ V
mpofvexi.3 𝑅 ∈ V
mpofvexi.4 𝑆 ∈ V
Assertion
Ref Expression
mpofvexi (𝑅𝐹𝑆) ∈ V
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝑆(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem mpofvexi
StepHypRef Expression
1 fnmpoi.2 . . 3 𝐶 ∈ V
21gen2 1450 . 2 𝑥𝑦 𝐶 ∈ V
3 mpofvexi.3 . 2 𝑅 ∈ V
4 mpofvexi.4 . 2 𝑆 ∈ V
5 fmpo.1 . . 3 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
65mpofvex 6203 . 2 ((∀𝑥𝑦 𝐶 ∈ V ∧ 𝑅 ∈ V ∧ 𝑆 ∈ V) → (𝑅𝐹𝑆) ∈ V)
72, 3, 4, 6mp3an 1337 1 (𝑅𝐹𝑆) ∈ V
Colors of variables: wff set class
Syntax hints:  wal 1351   = wceq 1353  wcel 2148  Vcvv 2737  (class class class)co 5874  cmpo 5876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-fo 5222  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141
This theorem is referenced by: (None)
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