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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 6711. (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 7162 | . . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
2 | 1 | neeq1i 3083 | . 2 ⊢ ((𝐴𝐹𝐵) ≠ ∅ ↔ (𝐹‘〈𝐴, 𝐵〉) ≠ ∅) |
3 | fvfundmfvn0 6711 | . 2 ⊢ ((𝐹‘〈𝐴, 𝐵〉) ≠ ∅ → (〈𝐴, 𝐵〉 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {〈𝐴, 𝐵〉}))) | |
4 | 2, 3 | sylbi 219 | 1 ⊢ ((𝐴𝐹𝐵) ≠ ∅ → (〈𝐴, 𝐵〉 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {〈𝐴, 𝐵〉}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2113 ≠ wne 3019 ∅c0 4294 {csn 4570 〈cop 4576 dom cdm 5558 ↾ cres 5560 Fun wfun 6352 ‘cfv 6358 (class class class)co 7159 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-res 5570 df-iota 6317 df-fun 6360 df-fv 6366 df-ov 7162 |
This theorem is referenced by: (None) |
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