Theorem ffnaov 47149

Description: An operation maps to a class to which all values belong, analogous to ffnov 7559. (Contributed by Alexander van der Vekens, 26-May-2017.)

Assertion

Ref	Expression
ffnaov	⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝑥𝐹𝑦)) ∈ 𝐶))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝐹,𝑦

Proof of Theorem ffnaov

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ffnafv 47121	. 2 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑤 ∈ (𝐴 × 𝐵)(𝐹'''𝑤) ∈ 𝐶))
2		afveq2 47085	. . . . . 6 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐹'''𝑤) = (𝐹'''⟨𝑥, 𝑦⟩))
3		df-aov 47071	. . . . . 6 ⊢ ((𝑥𝐹𝑦)) = (𝐹'''⟨𝑥, 𝑦⟩)
4	2, 3	eqtr4di 2793	. . . . 5 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐹'''𝑤) = ((𝑥𝐹𝑦)) )
5	4	eleq1d 2824	. . . 4 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → ((𝐹'''𝑤) ∈ 𝐶 ↔ ((𝑥𝐹𝑦)) ∈ 𝐶))
6	5	ralxp 5855	. . 3 ⊢ (∀𝑤 ∈ (𝐴 × 𝐵)(𝐹'''𝑤) ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝑥𝐹𝑦)) ∈ 𝐶)
7	6	anbi2i 623	. 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑤 ∈ (𝐴 × 𝐵)(𝐹'''𝑤) ∈ 𝐶) ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝑥𝐹𝑦)) ∈ 𝐶))
8	1, 7	bitri 275	1 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝑥𝐹𝑦)) ∈ 𝐶))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059  ⟨cop 4637   × cxp 5687   Fn wfn 6558  ⟶wf 6559  '''cafv 47067   ((caov 47068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-aiota 47035  df-dfat 47069  df-afv 47070  df-aov 47071

This theorem is referenced by:  faovcl  47150