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Theorem ovmpt2elrn 7104
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 7100 . 2 (∀𝑥𝐴𝑦𝐵 𝐶𝑀𝑂:(𝐴 × 𝐵)⟶𝑀)
3 fovrn 6676 . 2 ((𝑂:(𝐴 × 𝐵)⟶𝑀𝑋𝐴𝑌𝐵) → (𝑋𝑂𝑌) ∈ 𝑀)
42, 3syl3an1b 1353 1 ((∀𝑥𝐴𝑦𝐵 𝐶𝑀𝑋𝐴𝑌𝐵) → (𝑋𝑂𝑌) ∈ 𝑀)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1030   = wceq 1474  wcel 1976  wral 2892   × cxp 5023  wf 5783  (class class class)co 6524  cmpt2 6526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-ral 2897  df-rex 2898  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-sn 4122  df-pr 4124  df-op 4128  df-uni 4364  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-id 4940  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-fv 5795  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-1st 7033  df-2nd 7034
This theorem is referenced by:  opifismgm  17024  opmpt2ismgm  41596
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