Theorem mgcf1 33550

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 33549	. . 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 2145  ∀wral 3077   class class class wbr 5103  ⟶wf 6535  ‘cfv 6539  (class class class)co 7421  Basecbs 17388  lecple 17436   Proset cproset 18467  MGalConncmgc 33541

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7752

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3900  df-un 3902  df-in 3904  df-ss 3915  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-iota 6495  df-fun 6541  df-fn 6542  df-f 6543  df-fv 6547  df-ov 7424  df-oprab 7425  df-mpo 7426  df-map 8850  df-mgc 33543

This theorem is used by:  mgcmntco  33556  mgcmnt1d  33559