Theorem ovn0ssdmfun 7585

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 6922. (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 6882	. . . . 5 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝐹‘𝑝) = (𝐹‘⟨𝑎, 𝑏⟩))
2		df-ov 7419	. . . . 5 ⊢ (𝑎𝐹𝑏) = (𝐹‘⟨𝑎, 𝑏⟩)
3	1, 2	eqtr4di 2815	. . . 4 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝐹‘𝑝) = (𝑎𝐹𝑏))
4	3	neeq1d 3016	. . 3 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → ((𝐹‘𝑝) ≠ ∅ ↔ (𝑎𝐹𝑏) ≠ ∅))
5	4	ralxp 5825	. 2 ⊢ (∀𝑝 ∈ (𝐷 × 𝐸)(𝐹‘𝑝) ≠ ∅ ↔ ∀𝑎 ∈ 𝐷 ∀𝑏 ∈ 𝐸 (𝑎𝐹𝑏) ≠ ∅)
6		fvn0ssdmfun 7070	. 2 ⊢ (∀𝑝 ∈ (𝐷 × 𝐸)(𝐹‘𝑝) ≠ ∅ → ((𝐷 × 𝐸) ⊆ dom 𝐹 ∧ Fun (𝐹 ↾ (𝐷 × 𝐸))))
7	5, 6	sylbir 238	1 ⊢ (∀𝑎 ∈ 𝐷 ∀𝑏 ∈ 𝐸 (𝑎𝐹𝑏) ≠ ∅ → ((𝐷 × 𝐸) ⊆ dom 𝐹 ∧ Fun (𝐹 ↾ (𝐷 × 𝐸))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ≠ wne 2957  ∀wral 3078   ⊆ wss 3902  ∅c0 4282  ⟨cop 4593   × cxp 5657  dom cdm 5659   ↾ cres 5661  Fun wfun 6531  ‘cfv 6537  (class class class)co 7416

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-res 5671  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7419

This theorem is used by:  mgmn0plusgf  18743