Theorem ffnaov 41279

Description: An operation maps to a class to which all values belong, analogous to ffnov 6764. (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 41251	. 2 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑤 ∈ (𝐴 × 𝐵)(𝐹'''𝑤) ∈ 𝐶))
2		afveq2 41215	. . . . . 6 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐹'''𝑤) = (𝐹'''⟨𝑥, 𝑦⟩))
3		df-aov 41198	. . . . . 6 ⊢ ((𝑥𝐹𝑦)) = (𝐹'''⟨𝑥, 𝑦⟩)
4	2, 3	syl6eqr 2674	. . . . 5 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐹'''𝑤) = ((𝑥𝐹𝑦)) )
5	4	eleq1d 2686	. . . 4 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → ((𝐹'''𝑤) ∈ 𝐶 ↔ ((𝑥𝐹𝑦)) ∈ 𝐶))
6	5	ralxp 5263	. . 3 ⊢ (∀𝑤 ∈ (𝐴 × 𝐵)(𝐹'''𝑤) ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝑥𝐹𝑦)) ∈ 𝐶)
7	6	anbi2i 730	. 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 264	1 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝑥𝐹𝑦)) ∈ 𝐶))

Syntax hints:   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ⟨cop 4183   × cxp 5112   Fn wfn 5883  ⟶wf 5884  '''cafv 41194   ((caov 41195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-dfat 41196  df-afv 41197  df-aov 41198

This theorem is referenced by:  faovcl  41280