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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mgcf2 | Structured version Visualization version GIF version |
Description: The upper adjoint 𝐺 of a Galois connection is a function. (Contributed by Thierry Arnoux, 24-Apr-2024.) |
Ref | Expression |
---|---|
mgcoval.1 | ⊢ 𝐴 = (Base‘𝑉) |
mgcoval.2 | ⊢ 𝐵 = (Base‘𝑊) |
mgcoval.3 | ⊢ ≤ = (le‘𝑉) |
mgcoval.4 | ⊢ ≲ = (le‘𝑊) |
mgcval.1 | ⊢ 𝐻 = (𝑉MGalConn𝑊) |
mgcval.2 | ⊢ (𝜑 → 𝑉 ∈ Proset ) |
mgcval.3 | ⊢ (𝜑 → 𝑊 ∈ Proset ) |
mgccole.1 | ⊢ (𝜑 → 𝐹𝐻𝐺) |
Ref | Expression |
---|---|
mgcf2 | ⊢ (𝜑 → 𝐺:𝐵⟶𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mgccole.1 | . . 3 ⊢ (𝜑 → 𝐹𝐻𝐺) | |
2 | mgcoval.1 | . . . 4 ⊢ 𝐴 = (Base‘𝑉) | |
3 | mgcoval.2 | . . . 4 ⊢ 𝐵 = (Base‘𝑊) | |
4 | mgcoval.3 | . . . 4 ⊢ ≤ = (le‘𝑉) | |
5 | mgcoval.4 | . . . 4 ⊢ ≲ = (le‘𝑊) | |
6 | mgcval.1 | . . . 4 ⊢ 𝐻 = (𝑉MGalConn𝑊) | |
7 | mgcval.2 | . . . 4 ⊢ (𝜑 → 𝑉 ∈ Proset ) | |
8 | mgcval.3 | . . . 4 ⊢ (𝜑 → 𝑊 ∈ Proset ) | |
9 | 2, 3, 4, 5, 6, 7, 8 | mgcval 32028 | . . 3 ⊢ (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦))))) |
10 | 1, 9 | mpbid 231 | . 2 ⊢ (𝜑 → ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)))) |
11 | 10 | simplrd 768 | 1 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3060 class class class wbr 5141 ⟶wf 6528 ‘cfv 6532 (class class class)co 7393 Basecbs 17126 lecple 17186 Proset cproset 18228 MGalConncmgc 32020 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-fv 6540 df-ov 7396 df-oprab 7397 df-mpo 7398 df-map 8805 df-mgc 32022 |
This theorem is referenced by: mgcmntco 32035 mgcmnt2d 32039 |
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