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Theorem fovcld 30377
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 1145 . 2 ((𝜑𝐴𝑅𝐵𝑆) → (𝐴𝑅𝐵𝑆))
2 fovcld.1 . . . 4 (𝜑𝐹:(𝑅 × 𝑆)⟶𝐶)
3 ffnov 7270 . . . . 5 (𝐹:(𝑅 × 𝑆)⟶𝐶 ↔ (𝐹 Fn (𝑅 × 𝑆) ∧ ∀𝑥𝑅𝑦𝑆 (𝑥𝐹𝑦) ∈ 𝐶))
43simprbi 499 . . . 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 7155 . . . 4 (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦))
87eleq1d 2895 . . 3 (𝑥 = 𝐴 → ((𝑥𝐹𝑦) ∈ 𝐶 ↔ (𝐴𝐹𝑦) ∈ 𝐶))
9 oveq2 7156 . . . 4 (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵))
109eleq1d 2895 . . 3 (𝑦 = 𝐵 → ((𝐴𝐹𝑦) ∈ 𝐶 ↔ (𝐴𝐹𝐵) ∈ 𝐶))
118, 10rspc2v 3631 . 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 398  w3a 1082   = wceq 1531  wcel 2108  wral 3136   × cxp 5546   Fn wfn 6343  wf 6344  (class class class)co 7148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7151
This theorem is referenced by:  imaslmod  30915
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