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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mgcf1 | Structured version Visualization version GIF version |
Description: The lower adjoint 𝐹 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 |
---|---|
mgcf1 | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
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 30776 | . . 3 ⊢ (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦))))) |
10 | 1, 9 | mpbid 235 | . 2 ⊢ (𝜑 → ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)))) |
11 | 10 | simplld 768 | 1 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1539 ∈ wcel 2112 ∀wral 3068 class class class wbr 5025 ⟶wf 6324 ‘cfv 6328 (class class class)co 7143 Basecbs 16526 lecple 16615 Proset cproset 17587 MGalConncmgc 30768 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3an 1087 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ral 3073 df-rex 3074 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-op 4522 df-uni 4792 df-br 5026 df-opab 5088 df-id 5423 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-fv 6336 df-ov 7146 df-oprab 7147 df-mpo 7148 df-map 8411 df-mgc 30770 |
This theorem is referenced by: mgcmntco 30783 mgcmnt1d 30786 |
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