Theorem ovn0ssdmfun 48952

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 6921. (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 6881	. . . . 5 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝐹‘𝑝) = (𝐹‘⟨𝑎, 𝑏⟩))
2		df-ov 7413	. . . . 5 ⊢ (𝑎𝐹𝑏) = (𝐹‘⟨𝑎, 𝑏⟩)
3	1, 2	eqtr4di 2816	. . . 4 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝐹‘𝑝) = (𝑎𝐹𝑏))
4	3	neeq1d 3017	. . 3 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → ((𝐹‘𝑝) ≠ ∅ ↔ (𝑎𝐹𝑏) ≠ ∅))
5	4	ralxp 5827	. 2 ⊢ (∀𝑝 ∈ (𝐷 × 𝐸)(𝐹‘𝑝) ≠ ∅ ↔ ∀𝑎 ∈ 𝐷 ∀𝑏 ∈ 𝐸 (𝑎𝐹𝑏) ≠ ∅)
6		fvn0ssdmfun 7069	. 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 400   = wceq 1570   ≠ wne 2958  ∀wral 3079   ⊆ wss 3905  ∅c0 4286  ⟨cop 4595   × cxp 5659  dom cdm 5661   ↾ cres 5663  Fun wfun 6530  ‘cfv 6536  (class class class)co 7410

This proof depends on 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 proof 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-iun 4958  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-res 5673  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7413

This theorem is used by: (None)