ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fovrnd GIF version

Theorem fovrnd 5883
Description: An operation's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)
Hypotheses
Ref Expression
fovrnd.1 (𝜑𝐹:(𝑅 × 𝑆)⟶𝐶)
fovrnd.2 (𝜑𝐴𝑅)
fovrnd.3 (𝜑𝐵𝑆)
Assertion
Ref Expression
fovrnd (𝜑 → (𝐴𝐹𝐵) ∈ 𝐶)

Proof of Theorem fovrnd
StepHypRef Expression
1 fovrnd.1 . 2 (𝜑𝐹:(𝑅 × 𝑆)⟶𝐶)
2 fovrnd.2 . 2 (𝜑𝐴𝑅)
3 fovrnd.3 . 2 (𝜑𝐵𝑆)
4 fovrn 5881 . 2 ((𝐹:(𝑅 × 𝑆)⟶𝐶𝐴𝑅𝐵𝑆) → (𝐴𝐹𝐵) ∈ 𝐶)
51, 2, 3, 4syl3anc 1201 1 (𝜑 → (𝐴𝐹𝐵) ∈ 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1465   × cxp 4507  wf 5089  (class class class)co 5742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101  df-ov 5745
This theorem is referenced by:  eroveu  6488  isxmet2d  12444  ismet2  12450  comet  12595  bdmetval  12596  txmetcnp  12614  limccnp2lem  12741  limccnp2cntop  12742
  Copyright terms: Public domain W3C validator