MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ovmpoelrn Structured version   Visualization version   GIF version

Theorem ovmpoelrn 8084
Description: An operation's value belongs to its range. (Contributed by AV, 27-Jan-2020.)
Hypothesis
Ref Expression
ovmpoelrn.o 𝑂 = (𝑥𝐴, 𝑦𝐵𝐶)
Assertion
Ref Expression
ovmpoelrn ((∀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 ovmpoelrn
StepHypRef Expression
1 ovmpoelrn.o . . 3 𝑂 = (𝑥𝐴, 𝑦𝐵𝐶)
21fmpo 8080 . 2 (∀𝑥𝐴𝑦𝐵 𝐶𝑀𝑂:(𝐴 × 𝐵)⟶𝑀)
3 fovcdm 7598 . 2 ((𝑂:(𝐴 × 𝐵)⟶𝑀𝑋𝐴𝑌𝐵) → (𝑋𝑂𝑌) ∈ 𝑀)
42, 3syl3an1b 1400 1 ((∀𝑥𝐴𝑦𝐵 𝐶𝑀𝑋𝐴𝑌𝐵) → (𝑋𝑂𝑌) ∈ 𝑀)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1533  wcel 2098  wral 3058   × cxp 5680  wf 6549  (class class class)co 7426  cmpo 7428
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 8001  df-2nd 8002
This theorem is referenced by:  opifismgm  18628  opmpoismgm  47325
  Copyright terms: Public domain W3C validator