Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  faovcl Structured version   Visualization version   GIF version

Theorem faovcl 43771
 Description: Closure law for an operation, analogous to fovcl 7259. (Contributed by Alexander van der Vekens, 26-May-2017.)
Hypothesis
Ref Expression
faovcl.1 𝐹:(𝑅 × 𝑆)⟶𝐶
Assertion
Ref Expression
faovcl ((𝐴𝑅𝐵𝑆) → ((𝐴𝐹𝐵)) ∈ 𝐶)

Proof of Theorem faovcl
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 faovcl.1 . . 3 𝐹:(𝑅 × 𝑆)⟶𝐶
2 ffnaov 43770 . . . 4 (𝐹:(𝑅 × 𝑆)⟶𝐶 ↔ (𝐹 Fn (𝑅 × 𝑆) ∧ ∀𝑥𝑅𝑦𝑆 ((𝑥𝐹𝑦)) ∈ 𝐶))
32simprbi 500 . . 3 (𝐹:(𝑅 × 𝑆)⟶𝐶 → ∀𝑥𝑅𝑦𝑆 ((𝑥𝐹𝑦)) ∈ 𝐶)
41, 3ax-mp 5 . 2 𝑥𝑅𝑦𝑆 ((𝑥𝐹𝑦)) ∈ 𝐶
5 eqidd 2799 . . . . 5 (𝑥 = 𝐴𝐹 = 𝐹)
6 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
7 eqidd 2799 . . . . 5 (𝑥 = 𝐴𝑦 = 𝑦)
85, 6, 7aoveq123d 43749 . . . 4 (𝑥 = 𝐴 → ((𝑥𝐹𝑦)) = ((𝐴𝐹𝑦)) )
98eleq1d 2874 . . 3 (𝑥 = 𝐴 → ( ((𝑥𝐹𝑦)) ∈ 𝐶 ↔ ((𝐴𝐹𝑦)) ∈ 𝐶))
10 eqidd 2799 . . . . 5 (𝑦 = 𝐵𝐹 = 𝐹)
11 eqidd 2799 . . . . 5 (𝑦 = 𝐵𝐴 = 𝐴)
12 id 22 . . . . 5 (𝑦 = 𝐵𝑦 = 𝐵)
1310, 11, 12aoveq123d 43749 . . . 4 (𝑦 = 𝐵 → ((𝐴𝐹𝑦)) = ((𝐴𝐹𝐵)) )
1413eleq1d 2874 . . 3 (𝑦 = 𝐵 → ( ((𝐴𝐹𝑦)) ∈ 𝐶 ↔ ((𝐴𝐹𝐵)) ∈ 𝐶))
159, 14rspc2v 3581 . 2 ((𝐴𝑅𝐵𝑆) → (∀𝑥𝑅𝑦𝑆 ((𝑥𝐹𝑦)) ∈ 𝐶 → ((𝐴𝐹𝐵)) ∈ 𝐶))
164, 15mpi 20 1 ((𝐴𝑅𝐵𝑆) → ((𝐴𝐹𝐵)) ∈ 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106   × cxp 5517   Fn wfn 6319  ⟶wf 6320   ((caov 43689 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-aiota 43657  df-dfat 43690  df-afv 43691  df-aov 43692 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator