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Theorem fovcld 29747
Description: Closure law for an operation. (Contributed by NM, 19-Apr-2007.) (Revised by Thierry Arnoux, 17-Feb-2017.)
Hypothesis
Ref Expression
fovcld.1 (𝜑𝐹:(𝑅 × 𝑆)⟶𝐶)
Assertion
Ref Expression
fovcld ((𝜑𝐴𝑅𝐵𝑆) → (𝐴𝐹𝐵) ∈ 𝐶)

Proof of Theorem fovcld
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 3simpc 1147 . 2 ((𝜑𝐴𝑅𝐵𝑆) → (𝐴𝑅𝐵𝑆))
2 fovcld.1 . . . 4 (𝜑𝐹:(𝑅 × 𝑆)⟶𝐶)
3 ffnov 6927 . . . . 5 (𝐹:(𝑅 × 𝑆)⟶𝐶 ↔ (𝐹 Fn (𝑅 × 𝑆) ∧ ∀𝑥𝑅𝑦𝑆 (𝑥𝐹𝑦) ∈ 𝐶))
43simprbi 483 . . . 4 (𝐹:(𝑅 × 𝑆)⟶𝐶 → ∀𝑥𝑅𝑦𝑆 (𝑥𝐹𝑦) ∈ 𝐶)
52, 4syl 17 . . 3 (𝜑 → ∀𝑥𝑅𝑦𝑆 (𝑥𝐹𝑦) ∈ 𝐶)
653ad2ant1 1128 . 2 ((𝜑𝐴𝑅𝐵𝑆) → ∀𝑥𝑅𝑦𝑆 (𝑥𝐹𝑦) ∈ 𝐶)
7 oveq1 6818 . . . 4 (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦))
87eleq1d 2822 . . 3 (𝑥 = 𝐴 → ((𝑥𝐹𝑦) ∈ 𝐶 ↔ (𝐴𝐹𝑦) ∈ 𝐶))
9 oveq2 6819 . . . 4 (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵))
109eleq1d 2822 . . 3 (𝑦 = 𝐵 → ((𝐴𝐹𝑦) ∈ 𝐶 ↔ (𝐴𝐹𝐵) ∈ 𝐶))
118, 10rspc2v 3459 . 2 ((𝐴𝑅𝐵𝑆) → (∀𝑥𝑅𝑦𝑆 (𝑥𝐹𝑦) ∈ 𝐶 → (𝐴𝐹𝐵) ∈ 𝐶))
121, 6, 11sylc 65 1 ((𝜑𝐴𝑅𝐵𝑆) → (𝐴𝐹𝐵) ∈ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1630  wcel 2137  wral 3048   × cxp 5262   Fn wfn 6042  wf 6043  (class class class)co 6811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pr 5053
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ral 3053  df-rex 3054  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-id 5172  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-fv 6055  df-ov 6814
This theorem is referenced by: (None)
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