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Mirrors > Home > MPE Home > Th. List > Mathboxes > ovn0dmfun | Structured version Visualization version GIF version |
Description: If a class operation value for two operands is not the empty set, then the operands are contained in the domain of the class, and the class restricted to the operands is a function, analogous to fvfundmfvn0 6383. (Contributed by AV, 27-Jan-2020.) |
Ref | Expression |
---|---|
ovn0dmfun | ⊢ ((𝐴𝐹𝐵) ≠ ∅ → (〈𝐴, 𝐵〉 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {〈𝐴, 𝐵〉}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ov 6812 | . . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
2 | 1 | neeq1i 2992 | . 2 ⊢ ((𝐴𝐹𝐵) ≠ ∅ ↔ (𝐹‘〈𝐴, 𝐵〉) ≠ ∅) |
3 | fvfundmfvn0 6383 | . 2 ⊢ ((𝐹‘〈𝐴, 𝐵〉) ≠ ∅ → (〈𝐴, 𝐵〉 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {〈𝐴, 𝐵〉}))) | |
4 | 2, 3 | sylbi 207 | 1 ⊢ ((𝐴𝐹𝐵) ≠ ∅ → (〈𝐴, 𝐵〉 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {〈𝐴, 𝐵〉}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 2135 ≠ wne 2928 ∅c0 4054 {csn 4317 〈cop 4323 dom cdm 5262 ↾ cres 5264 Fun wfun 6039 ‘cfv 6045 (class class class)co 6809 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1867 ax-4 1882 ax-5 1984 ax-6 2050 ax-7 2086 ax-8 2137 ax-9 2144 ax-10 2164 ax-11 2179 ax-12 2192 ax-13 2387 ax-ext 2736 ax-sep 4929 ax-nul 4937 ax-pow 4988 ax-pr 5051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1631 df-ex 1850 df-nf 1855 df-sb 2043 df-eu 2607 df-mo 2608 df-clab 2743 df-cleq 2749 df-clel 2752 df-nfc 2887 df-ne 2929 df-ral 3051 df-rex 3052 df-rab 3055 df-v 3338 df-dif 3714 df-un 3716 df-in 3718 df-ss 3725 df-nul 4055 df-if 4227 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4585 df-br 4801 df-opab 4861 df-id 5170 df-xp 5268 df-rel 5269 df-cnv 5270 df-co 5271 df-dm 5272 df-res 5274 df-iota 6008 df-fun 6047 df-fv 6053 df-ov 6812 |
This theorem is referenced by: (None) |
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