Theorem mgcf1 33616

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

Assertion

Ref	Expression
mgcf1	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)

Proof of Theorem mgcf1

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 33615	. . 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 780	1 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080   class class class wbr 5106  ⟶wf 6551  ‘cfv 6555  (class class class)co 7440  Basecbs 17413  lecple 17461   Proset cproset 18492  MGalConncmgc 33607

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-fv 6563  df-ov 7443  df-oprab 7444  df-mpo 7445  df-map 8874  df-mgc 33609

This theorem is used by:  mgcmntco  33622  mgcmnt1d  33625