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Theorem mpt2fvexi 5859
Description: Sufficient condition for an operation maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)
Hypotheses
Ref Expression
fmpt2.1 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
fnmpt2i.2 𝐶 ∈ V
mpt2fvexi.3 𝑅 ∈ V
mpt2fvexi.4 𝑆 ∈ V
Assertion
Ref Expression
mpt2fvexi (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 mpt2fvexi
StepHypRef Expression
1 fnmpt2i.2 . . 3 𝐶 ∈ V
21gen2 1355 . 2 𝑥𝑦 𝐶 ∈ V
3 mpt2fvexi.3 . 2 𝑅 ∈ V
4 mpt2fvexi.4 . 2 𝑆 ∈ V
5 fmpt2.1 . . 3 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
65mpt2fvex 5856 . 2 ((∀𝑥𝑦 𝐶 ∈ V ∧ 𝑅 ∈ V ∧ 𝑆 ∈ V) → (𝑅𝐹𝑆) ∈ V)
72, 3, 4, 6mp3an 1243 1 (𝑅𝐹𝑆) ∈ V
Colors of variables: wff set class
Syntax hints:  wal 1257   = wceq 1259  wcel 1409  Vcvv 2574  (class class class)co 5539  cmpt2 5541
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2787  df-csb 2880  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-fo 4935  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795
This theorem is referenced by: (None)
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