Theorem mgcf2 33067

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 33065	. . 3 ⊢ (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)))))
10	1, 9	mpbid 232	. 2 ⊢ (𝜑 → ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦))))
11	10	simplrd 770	1 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052   class class class wbr 5086  ⟶wf 6489  ‘cfv 6493  (class class class)co 7361  Basecbs 17173  lecple 17221   Proset cproset 18252  MGalConncmgc 33057

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-fv 6501  df-ov 7364  df-oprab 7365  df-mpo 7366  df-map 8769  df-mgc 33059

This theorem is referenced by:  mgcmntco  33072  mgcmnt2d  33076