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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 6226. (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 6191 | . . . . 5 ⊢ (𝑝 = 〈𝑎, 𝑏〉 → (𝐹‘𝑝) = (𝐹‘〈𝑎, 𝑏〉)) | |
2 | df-ov 6653 | . . . . 5 ⊢ (𝑎𝐹𝑏) = (𝐹‘〈𝑎, 𝑏〉) | |
3 | 1, 2 | syl6eqr 2674 | . . . 4 ⊢ (𝑝 = 〈𝑎, 𝑏〉 → (𝐹‘𝑝) = (𝑎𝐹𝑏)) |
4 | 3 | neeq1d 2853 | . . 3 ⊢ (𝑝 = 〈𝑎, 𝑏〉 → ((𝐹‘𝑝) ≠ ∅ ↔ (𝑎𝐹𝑏) ≠ ∅)) |
5 | 4 | ralxp 5263 | . 2 ⊢ (∀𝑝 ∈ (𝐷 × 𝐸)(𝐹‘𝑝) ≠ ∅ ↔ ∀𝑎 ∈ 𝐷 ∀𝑏 ∈ 𝐸 (𝑎𝐹𝑏) ≠ ∅) |
6 | fvn0ssdmfun 6350 | . 2 ⊢ (∀𝑝 ∈ (𝐷 × 𝐸)(𝐹‘𝑝) ≠ ∅ → ((𝐷 × 𝐸) ⊆ dom 𝐹 ∧ Fun (𝐹 ↾ (𝐷 × 𝐸)))) | |
7 | 5, 6 | sylbir 225 | 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 384 = wceq 1483 ≠ wne 2794 ∀wral 2912 ⊆ wss 3574 ∅c0 3915 〈cop 4183 × cxp 5112 dom cdm 5114 ↾ cres 5116 Fun wfun 5882 ‘cfv 5888 (class class class)co 6650 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-res 5126 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 |
This theorem is referenced by: (None) |
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