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Theorem ovn0ssdmfun 42285
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 6367. (Contributed by AV, 27-Jan-2020.)
Assertion
Ref Expression
ovn0ssdmfun (∀𝑎𝐷𝑏𝐸 (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.
StepHypRef Expression
1 fveq2 6332 . . . . 5 (𝑝 = ⟨𝑎, 𝑏⟩ → (𝐹𝑝) = (𝐹‘⟨𝑎, 𝑏⟩))
2 df-ov 6795 . . . . 5 (𝑎𝐹𝑏) = (𝐹‘⟨𝑎, 𝑏⟩)
31, 2syl6eqr 2822 . . . 4 (𝑝 = ⟨𝑎, 𝑏⟩ → (𝐹𝑝) = (𝑎𝐹𝑏))
43neeq1d 3001 . . 3 (𝑝 = ⟨𝑎, 𝑏⟩ → ((𝐹𝑝) ≠ ∅ ↔ (𝑎𝐹𝑏) ≠ ∅))
54ralxp 5402 . 2 (∀𝑝 ∈ (𝐷 × 𝐸)(𝐹𝑝) ≠ ∅ ↔ ∀𝑎𝐷𝑏𝐸 (𝑎𝐹𝑏) ≠ ∅)
6 fvn0ssdmfun 6493 . 2 (∀𝑝 ∈ (𝐷 × 𝐸)(𝐹𝑝) ≠ ∅ → ((𝐷 × 𝐸) ⊆ dom 𝐹 ∧ Fun (𝐹 ↾ (𝐷 × 𝐸))))
75, 6sylbir 225 1 (∀𝑎𝐷𝑏𝐸 (𝑎𝐹𝑏) ≠ ∅ → ((𝐷 × 𝐸) ⊆ dom 𝐹 ∧ Fun (𝐹 ↾ (𝐷 × 𝐸))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1630  wne 2942  wral 3060  wss 3721  c0 4061  cop 4320   × cxp 5247  dom cdm 5249  cres 5251  Fun wfun 6025  cfv 6031  (class class class)co 6792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-res 5261  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6795
This theorem is referenced by: (None)
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