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

Theorem ovmpoelrn 8057
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 8053 . 2 (∀𝑥𝐴𝑦𝐵 𝐶𝑀𝑂:(𝐴 × 𝐵)⟶𝑀)
3 fovcdm 7574 . 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 3055   × cxp 5667  wf 6533  (class class class)co 7405  cmpo 7407
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975
This theorem is referenced by:  opifismgm  18592  opmpoismgm  47117
  Copyright terms: Public domain W3C validator