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Theorem mpt2fvexi 5832
 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 1339 . 2 𝑥𝑦 𝐶 ∈ V
3 mpt2fvexi.3 . 2 𝑅 ∈ V
4 mpt2fvexi.4 . 2 𝑆 ∈ V
5 fmpt2.1 . . 3 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
65mpt2fvex 5829 . 2 ((∀𝑥𝑦 𝐶 ∈ V ∧ 𝑅 ∈ V ∧ 𝑆 ∈ V) → (𝑅𝐹𝑆) ∈ V)
72, 3, 4, 6mp3an 1232 1 (𝑅𝐹𝑆) ∈ V
 Colors of variables: wff set class Syntax hints:  ∀wal 1241   = wceq 1243   ∈ wcel 1393  Vcvv 2557  (class class class)co 5512   ↦ cmpt2 5514 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fo 4908  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768 This theorem is referenced by: (None)
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