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

Theorem fovrnd 5789
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 5787 . 2 ((𝐹:(𝑅 × 𝑆)⟶𝐶𝐴𝑅𝐵𝑆) → (𝐴𝐹𝐵) ∈ 𝐶)
51, 2, 3, 4syl3anc 1174 1 (𝜑 → (𝐴𝐹𝐵) ∈ 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1438   × cxp 4436  wf 5011  (class class class)co 5652
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-fv 5023  df-ov 5655
This theorem is referenced by:  eroveu  6383
  Copyright terms: Public domain W3C validator