Theorem ffnaov 47956

Description: An operation maps to a class to which all values belong, analogous to ffnov 7536. (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 47928	. 2 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑤 ∈ (𝐴 × 𝐵)(𝐹'''𝑤) ∈ 𝐶))
2		afveq2 47892	. . . . . 6 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐹'''𝑤) = (𝐹'''⟨𝑥, 𝑦⟩))
3		df-aov 47878	. . . . . 6 ⊢ ((𝑥𝐹𝑦)) = (𝐹'''⟨𝑥, 𝑦⟩)
4	2, 3	eqtr4di 2816	. . . . 5 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐹'''𝑤) = ((𝑥𝐹𝑦)) )
5	4	eleq1d 2848	. . . 4 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → ((𝐹'''𝑤) ∈ 𝐶 ↔ ((𝑥𝐹𝑦)) ∈ 𝐶))
6	5	ralxp 5827	. . 3 ⊢ (∀𝑤 ∈ (𝐴 × 𝐵)(𝐹'''𝑤) ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝑥𝐹𝑦)) ∈ 𝐶)
7	6	anbi2i 634	. 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 278	1 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝑥𝐹𝑦)) ∈ 𝐶))

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ⟨cop 4595   × cxp 5659   Fn wfn 6531  ⟶wf 6532  '''cafv 47874   ((caov 47875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-aiota 47842  df-dfat 47876  df-afv 47877  df-aov 47878

This theorem is referenced by:  faovcl  47957