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Theorem mgcf1 30792
 Description: The lower adjoint 𝐹 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
mgcf1 (𝜑𝐹:𝐴𝐵)

Proof of Theorem mgcf1
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 30791 . . 3 (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)))))
101, 9mpbid 235 . 2 (𝜑 → ((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦))))
1110simplld 767 1 (𝜑𝐹:𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070   class class class wbr 5032  ⟶wf 6331  ‘cfv 6335  (class class class)co 7150  Basecbs 16541  lecple 16630   Proset cproset 17602  MGalConncmgc 30783 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-fv 6343  df-ov 7153  df-oprab 7154  df-mpo 7155  df-map 8418  df-mgc 30785 This theorem is referenced by:  mgcmntco  30798  mgcmnt1d  30801
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