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Theorem ovmpt2elrn 7201
Description: An operation's value belongs to its range. (Contributed by AV, 27-Jan-2020.)
Hypothesis
Ref Expression
ovmpt2elrn.o 𝑂 = (𝑥𝐴, 𝑦𝐵𝐶)
Assertion
Ref Expression
ovmpt2elrn ((∀𝑥𝐴𝑦𝐵 𝐶𝑀𝑋𝐴𝑌𝐵) → (𝑋𝑂𝑌) ∈ 𝑀)
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑀,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)   𝑂(𝑥,𝑦)   𝑋(𝑥,𝑦)   𝑌(𝑥,𝑦)

Proof of Theorem ovmpt2elrn
StepHypRef Expression
1 ovmpt2elrn.o . . 3 𝑂 = (𝑥𝐴, 𝑦𝐵𝐶)
21fmpt2 7197 . 2 (∀𝑥𝐴𝑦𝐵 𝐶𝑀𝑂:(𝐴 × 𝐵)⟶𝑀)
3 fovrn 6769 . 2 ((𝑂:(𝐴 × 𝐵)⟶𝑀𝑋𝐴𝑌𝐵) → (𝑋𝑂𝑌) ∈ 𝑀)
42, 3syl3an1b 1359 1 ((∀𝑥𝐴𝑦𝐵 𝐶𝑀𝑋𝐴𝑌𝐵) → (𝑋𝑂𝑌) ∈ 𝑀)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036   = wceq 1480  wcel 1987  wral 2908   × cxp 5082  wf 5853  (class class class)co 6615  cmpt2 6617
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-1st 7128  df-2nd 7129
This theorem is referenced by:  opifismgm  17198  opmpt2ismgm  41125
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