Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ovn0ssdmfun Structured version   Visualization version   GIF version

Theorem ovn0ssdmfun 41559
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 6121. (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 6088 . . . . 5 (𝑝 = ⟨𝑎, 𝑏⟩ → (𝐹𝑝) = (𝐹‘⟨𝑎, 𝑏⟩))
2 df-ov 6530 . . . . 5 (𝑎𝐹𝑏) = (𝐹‘⟨𝑎, 𝑏⟩)
31, 2syl6eqr 2661 . . . 4 (𝑝 = ⟨𝑎, 𝑏⟩ → (𝐹𝑝) = (𝑎𝐹𝑏))
43neeq1d 2840 . . 3 (𝑝 = ⟨𝑎, 𝑏⟩ → ((𝐹𝑝) ≠ ∅ ↔ (𝑎𝐹𝑏) ≠ ∅))
54ralxp 5173 . 2 (∀𝑝 ∈ (𝐷 × 𝐸)(𝐹𝑝) ≠ ∅ ↔ ∀𝑎𝐷𝑏𝐸 (𝑎𝐹𝑏) ≠ ∅)
6 fvn0ssdmfun 6243 . 2 (∀𝑝 ∈ (𝐷 × 𝐸)(𝐹𝑝) ≠ ∅ → ((𝐷 × 𝐸) ⊆ dom 𝐹 ∧ Fun (𝐹 ↾ (𝐷 × 𝐸))))
75, 6sylbir 223 1 (∀𝑎𝐷𝑏𝐸 (𝑎𝐹𝑏) ≠ ∅ → ((𝐷 × 𝐸) ⊆ dom 𝐹 ∧ Fun (𝐹 ↾ (𝐷 × 𝐸))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wne 2779  wral 2895  wss 3539  c0 3873  cop 4130   × cxp 5026  dom cdm 5028  cres 5030  Fun wfun 5784  cfv 5790  (class class class)co 6527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-res 5040  df-iota 5754  df-fun 5792  df-fv 5798  df-ov 6530
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator