Theorem mgcf2 32159

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	⊢ (𝜑 → 𝑉 ∈ Proset )
mgcval.3	⊢ (𝜑 → 𝑊 ∈ Proset )
mgccole.1	⊢ (𝜑 → 𝐹𝐻𝐺)

Assertion

Ref	Expression
mgcf2	⊢ (𝜑 → 𝐺:𝐵⟶𝐴)

Proof of Theorem mgcf2

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

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 32157	. . 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 769	1 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∀wral 3062   class class class wbr 5149  ⟶wf 6540  ‘cfv 6544  (class class class)co 7409  Basecbs 17144  lecple 17204   Proset cproset 18246  MGalConncmgc 32149

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-map 8822  df-mgc 32151

This theorem is referenced by:  mgcmntco  32164  mgcmnt2d  32168