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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 6702. (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 6664 | . . . . 5 ⊢ (𝑝 = 〈𝑎, 𝑏〉 → (𝐹‘𝑝) = (𝐹‘〈𝑎, 𝑏〉)) | |
2 | df-ov 7153 | . . . . 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 5706 | . 2 ⊢ (∀𝑝 ∈ (𝐷 × 𝐸)(𝐹‘𝑝) ≠ ∅ ↔ ∀𝑎 ∈ 𝐷 ∀𝑏 ∈ 𝐸 (𝑎𝐹𝑏) ≠ ∅) |
6 | fvn0ssdmfun 6836 | . 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 1533 ≠ wne 3016 ∀wral 3138 ⊆ wss 3935 ∅c0 4290 〈cop 4566 × cxp 5547 dom cdm 5549 ↾ cres 5551 Fun wfun 6343 ‘cfv 6349 (class class class)co 7150 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-res 5561 df-iota 6308 df-fun 6351 df-fv 6357 df-ov 7153 |
This theorem is referenced by: (None) |
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