Theorem ovn0ssdmfun 41557

Description: If a class' operation value for two operands is not the empty set, the operands are contained in the domain of the class, and the class restricted to the operands is a function, analogous to fvfundmfvn0 6136. (Contributed by AV, 27-Jan-2020.)

Assertion

Ref	Expression
ovn0ssdmfun	⊢ (∀𝑎 ∈ 𝐷 ∀𝑏 ∈ 𝐸 (𝑎𝐹𝑏) ≠ ∅ → ((𝐷 × 𝐸) ⊆ dom 𝐹 ∧ Fun (𝐹 ↾ (𝐷 × 𝐸))))

Distinct variable groups:   𝐷,𝑎,𝑏   𝐸,𝑎,𝑏   𝐹,𝑎,𝑏

Proof of Theorem ovn0ssdmfun

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6103	. . . . 5 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝐹‘𝑝) = (𝐹‘⟨𝑎, 𝑏⟩))
2		df-ov 6552	. . . . 5 ⊢ (𝑎𝐹𝑏) = (𝐹‘⟨𝑎, 𝑏⟩)
3	1, 2	syl6eqr 2662	. . . 4 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝐹‘𝑝) = (𝑎𝐹𝑏))
4	3	neeq1d 2841	. . 3 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → ((𝐹‘𝑝) ≠ ∅ ↔ (𝑎𝐹𝑏) ≠ ∅))
5	4	ralxp 5185	. 2 ⊢ (∀𝑝 ∈ (𝐷 × 𝐸)(𝐹‘𝑝) ≠ ∅ ↔ ∀𝑎 ∈ 𝐷 ∀𝑏 ∈ 𝐸 (𝑎𝐹𝑏) ≠ ∅)
6		fvn0ssdmfun 6258	. 2 ⊢ (∀𝑝 ∈ (𝐷 × 𝐸)(𝐹‘𝑝) ≠ ∅ → ((𝐷 × 𝐸) ⊆ dom 𝐹 ∧ Fun (𝐹 ↾ (𝐷 × 𝐸))))
7	5, 6	sylbir 224	1 ⊢ (∀𝑎 ∈ 𝐷 ∀𝑏 ∈ 𝐸 (𝑎𝐹𝑏) ≠ ∅ → ((𝐷 × 𝐸) ⊆ dom 𝐹 ∧ Fun (𝐹 ↾ (𝐷 × 𝐸))))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ≠ wne 2780  ∀wral 2896   ⊆ wss 3540  ∅c0 3874  ⟨cop 4131   × cxp 5036  dom cdm 5038   ↾ cres 5040  Fun wfun 5798  ‘cfv 5804  (class class class)co 6549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-res 5050  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552

This theorem is referenced by: (None)