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Theorem mgcf2 32030
Description: The upper adjoint 𝐺 of a Galois connection is a function. (Contributed by Thierry Arnoux, 24-Apr-2024.)
Hypotheses
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 (𝜑𝑉 ∈ Proset )
mgcval.3 (𝜑𝑊 ∈ Proset )
mgccole.1 (𝜑𝐹𝐻𝐺)
Assertion
Ref Expression
mgcf2 (𝜑𝐺:𝐵𝐴)

Proof of Theorem mgcf2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mgccole.1 . . 3 (𝜑𝐹𝐻𝐺)
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 (𝜑𝑉 ∈ Proset )
8 mgcval.3 . . . 4 (𝜑𝑊 ∈ Proset )
92, 3, 4, 5, 6, 7, 8mgcval 32028 . . 3 (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)))))
101, 9mpbid 231 . 2 (𝜑 → ((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦))))
1110simplrd 768 1 (𝜑𝐺:𝐵𝐴)
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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