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Theorem fovcl 5928
Description: Closure law for an operation. (Contributed by NM, 19-Apr-2007.)
Hypothesis
Ref Expression
fovcl.1 𝐹:(𝑅 × 𝑆)⟶𝐶
Assertion
Ref Expression
fovcl ((𝐴𝑅𝐵𝑆) → (𝐴𝐹𝐵) ∈ 𝐶)

Proof of Theorem fovcl
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fovcl.1 . . 3 𝐹:(𝑅 × 𝑆)⟶𝐶
2 ffnov 5927 . . . 4 (𝐹:(𝑅 × 𝑆)⟶𝐶 ↔ (𝐹 Fn (𝑅 × 𝑆) ∧ ∀𝑥𝑅𝑦𝑆 (𝑥𝐹𝑦) ∈ 𝐶))
32simprbi 273 . . 3 (𝐹:(𝑅 × 𝑆)⟶𝐶 → ∀𝑥𝑅𝑦𝑆 (𝑥𝐹𝑦) ∈ 𝐶)
41, 3ax-mp 5 . 2 𝑥𝑅𝑦𝑆 (𝑥𝐹𝑦) ∈ 𝐶
5 oveq1 5833 . . . 4 (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦))
65eleq1d 2226 . . 3 (𝑥 = 𝐴 → ((𝑥𝐹𝑦) ∈ 𝐶 ↔ (𝐴𝐹𝑦) ∈ 𝐶))
7 oveq2 5834 . . . 4 (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵))
87eleq1d 2226 . . 3 (𝑦 = 𝐵 → ((𝐴𝐹𝑦) ∈ 𝐶 ↔ (𝐴𝐹𝐵) ∈ 𝐶))
96, 8rspc2v 2829 . 2 ((𝐴𝑅𝐵𝑆) → (∀𝑥𝑅𝑦𝑆 (𝑥𝐹𝑦) ∈ 𝐶 → (𝐴𝐹𝐵) ∈ 𝐶))
104, 9mpi 15 1 ((𝐴𝑅𝐵𝑆) → (𝐴𝐹𝐵) ∈ 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1335  wcel 2128  wral 2435   × cxp 4586   Fn wfn 5167  wf 5168  (class class class)co 5826
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-iun 3853  df-br 3968  df-opab 4028  df-mpt 4029  df-id 4255  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-iota 5137  df-fun 5174  df-fn 5175  df-f 5176  df-fv 5180  df-ov 5829
This theorem is referenced by:  xaddcl  9770  ixxssxr  9810  fzof  10052  elfzoelz  10055  fzoval  10056  addcncntoplem  13021
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