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Theorem mpt2fvexi 5958
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 1384 . 2 𝑥𝑦 𝐶 ∈ V
3 mpt2fvexi.3 . 2 𝑅 ∈ V
4 mpt2fvexi.4 . 2 𝑆 ∈ V
5 fmpt2.1 . . 3 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
65mpt2fvex 5955 . 2 ((∀𝑥𝑦 𝐶 ∈ V ∧ 𝑅 ∈ V ∧ 𝑆 ∈ V) → (𝑅𝐹𝑆) ∈ V)
72, 3, 4, 6mp3an 1273 1 (𝑅𝐹𝑆) ∈ V
Colors of variables: wff set class
Syntax hints:  wal 1287   = wceq 1289  wcel 1438  Vcvv 2619  (class class class)co 5634  cmpt2 5636
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-csb 2932  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-fo 5008  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894
This theorem is referenced by: (None)
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