Theorem ovn0ssdmfun 48147

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 6901. (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 6858	. . . . 5 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝐹‘𝑝) = (𝐹‘⟨𝑎, 𝑏⟩))
2		df-ov 7390	. . . . 5 ⊢ (𝑎𝐹𝑏) = (𝐹‘⟨𝑎, 𝑏⟩)
3	1, 2	eqtr4di 2782	. . . 4 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝐹‘𝑝) = (𝑎𝐹𝑏))
4	3	neeq1d 2984	. . 3 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → ((𝐹‘𝑝) ≠ ∅ ↔ (𝑎𝐹𝑏) ≠ ∅))
5	4	ralxp 5805	. 2 ⊢ (∀𝑝 ∈ (𝐷 × 𝐸)(𝐹‘𝑝) ≠ ∅ ↔ ∀𝑎 ∈ 𝐷 ∀𝑏 ∈ 𝐸 (𝑎𝐹𝑏) ≠ ∅)
6		fvn0ssdmfun 7046	. 2 ⊢ (∀𝑝 ∈ (𝐷 × 𝐸)(𝐹‘𝑝) ≠ ∅ → ((𝐷 × 𝐸) ⊆ dom 𝐹 ∧ Fun (𝐹 ↾ (𝐷 × 𝐸))))
7	5, 6	sylbir 235	1 ⊢ (∀𝑎 ∈ 𝐷 ∀𝑏 ∈ 𝐸 (𝑎𝐹𝑏) ≠ ∅ → ((𝐷 × 𝐸) ⊆ dom 𝐹 ∧ Fun (𝐹 ↾ (𝐷 × 𝐸))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ≠ wne 2925  ∀wral 3044   ⊆ wss 3914  ∅c0 4296  ⟨cop 4595   × cxp 5636  dom cdm 5638   ↾ cres 5640  Fun wfun 6505  ‘cfv 6511  (class class class)co 7387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-res 5650  df-iota 6464  df-fun 6513  df-fv 6519  df-ov 7390

This theorem is referenced by: (None)