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Mirrors > Home > MPE Home > Th. List > Mathboxes > ovn0ssdmfun | Structured version Visualization version GIF version |
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 6708. (Contributed by AV, 27-Jan-2020.) |
Ref | Expression |
---|---|
ovn0ssdmfun | ⊢ (∀𝑎 ∈ 𝐷 ∀𝑏 ∈ 𝐸 (𝑎𝐹𝑏) ≠ ∅ → ((𝐷 × 𝐸) ⊆ dom 𝐹 ∧ Fun (𝐹 ↾ (𝐷 × 𝐸)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6670 | . . . . 5 ⊢ (𝑝 = 〈𝑎, 𝑏〉 → (𝐹‘𝑝) = (𝐹‘〈𝑎, 𝑏〉)) | |
2 | df-ov 7159 | . . . . 5 ⊢ (𝑎𝐹𝑏) = (𝐹‘〈𝑎, 𝑏〉) | |
3 | 1, 2 | syl6eqr 2874 | . . . 4 ⊢ (𝑝 = 〈𝑎, 𝑏〉 → (𝐹‘𝑝) = (𝑎𝐹𝑏)) |
4 | 3 | neeq1d 3075 | . . 3 ⊢ (𝑝 = 〈𝑎, 𝑏〉 → ((𝐹‘𝑝) ≠ ∅ ↔ (𝑎𝐹𝑏) ≠ ∅)) |
5 | 4 | ralxp 5712 | . 2 ⊢ (∀𝑝 ∈ (𝐷 × 𝐸)(𝐹‘𝑝) ≠ ∅ ↔ ∀𝑎 ∈ 𝐷 ∀𝑏 ∈ 𝐸 (𝑎𝐹𝑏) ≠ ∅) |
6 | fvn0ssdmfun 6842 | . 2 ⊢ (∀𝑝 ∈ (𝐷 × 𝐸)(𝐹‘𝑝) ≠ ∅ → ((𝐷 × 𝐸) ⊆ dom 𝐹 ∧ Fun (𝐹 ↾ (𝐷 × 𝐸)))) | |
7 | 5, 6 | sylbir 237 | 1 ⊢ (∀𝑎 ∈ 𝐷 ∀𝑏 ∈ 𝐸 (𝑎𝐹𝑏) ≠ ∅ → ((𝐷 × 𝐸) ⊆ dom 𝐹 ∧ Fun (𝐹 ↾ (𝐷 × 𝐸)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ≠ wne 3016 ∀wral 3138 ⊆ wss 3936 ∅c0 4291 〈cop 4573 × cxp 5553 dom cdm 5555 ↾ cres 5557 Fun wfun 6349 ‘cfv 6355 (class class class)co 7156 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-res 5567 df-iota 6314 df-fun 6357 df-fv 6363 df-ov 7159 |
This theorem is referenced by: (None) |
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