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Theorem mgcval 30739
 Description: Monotone Galois connection between two functions 𝐹 and 𝐺. If this relation is satisfied, 𝐹 is called the lower adjoint of 𝐺, and 𝐺 is called the upper adjoint of 𝐹. Technically, this is implemented as an operation taking a pair of structures 𝑉 and 𝑊, expected to be posets, which gives a relation between pairs of functions 𝐹 and 𝐺. If such a relation exists, it can be proven to be unique. Galois connections generalize the fundamental theorem of Galois theory about the correspondence between subgroups and subfields. (Contributed by Thierry Arnoux, 23-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 )
Assertion
Ref Expression
mgcval (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)))))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑉,𝑦   𝑥,𝑊,𝑦   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐻(𝑥,𝑦)   (𝑥,𝑦)   (𝑥,𝑦)

Proof of Theorem mgcval
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mgcval.1 . . . 4 𝐻 = (𝑉MGalConn𝑊)
2 mgcval.2 . . . . 5 (𝜑𝑉 ∈ Proset )
3 mgcval.3 . . . . 5 (𝜑𝑊 ∈ Proset )
4 mgcoval.1 . . . . . 6 𝐴 = (Base‘𝑉)
5 mgcoval.2 . . . . . 6 𝐵 = (Base‘𝑊)
6 mgcoval.3 . . . . . 6 = (le‘𝑉)
7 mgcoval.4 . . . . . 6 = (le‘𝑊)
84, 5, 6, 7mgcoval 30738 . . . . 5 ((𝑉 ∈ Proset ∧ 𝑊 ∈ Proset ) → (𝑉MGalConn𝑊) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑔 ∈ (𝐴m 𝐵)) ∧ ∀𝑥𝐴𝑦𝐵 ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦)))})
92, 3, 8syl2anc 587 . . . 4 (𝜑 → (𝑉MGalConn𝑊) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑔 ∈ (𝐴m 𝐵)) ∧ ∀𝑥𝐴𝑦𝐵 ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦)))})
101, 9syl5eq 2845 . . 3 (𝜑𝐻 = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑔 ∈ (𝐴m 𝐵)) ∧ ∀𝑥𝐴𝑦𝐵 ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦)))})
1110breqd 5045 . 2 (𝜑 → (𝐹𝐻𝐺𝐹{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑔 ∈ (𝐴m 𝐵)) ∧ ∀𝑥𝐴𝑦𝐵 ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦)))}𝐺))
12 fveq1 6654 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
1312adantr 484 . . . . . . 7 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓𝑥) = (𝐹𝑥))
1413breq1d 5044 . . . . . 6 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑓𝑥) 𝑦 ↔ (𝐹𝑥) 𝑦))
15 fveq1 6654 . . . . . . . 8 (𝑔 = 𝐺 → (𝑔𝑦) = (𝐺𝑦))
1615adantl 485 . . . . . . 7 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑔𝑦) = (𝐺𝑦))
1716breq2d 5046 . . . . . 6 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑥 (𝑔𝑦) ↔ 𝑥 (𝐺𝑦)))
1814, 17bibi12d 349 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → (((𝑓𝑥) 𝑦𝑥 (𝑔𝑦)) ↔ ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦))))
19182ralbidv 3164 . . . 4 ((𝑓 = 𝐹𝑔 = 𝐺) → (∀𝑥𝐴𝑦𝐵 ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦)) ↔ ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦))))
20 eqid 2798 . . . 4 {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑔 ∈ (𝐴m 𝐵)) ∧ ∀𝑥𝐴𝑦𝐵 ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦)))} = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑔 ∈ (𝐴m 𝐵)) ∧ ∀𝑥𝐴𝑦𝐵 ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦)))}
2119, 20brab2a 5612 . . 3 (𝐹{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑔 ∈ (𝐴m 𝐵)) ∧ ∀𝑥𝐴𝑦𝐵 ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦)))}𝐺 ↔ ((𝐹 ∈ (𝐵m 𝐴) ∧ 𝐺 ∈ (𝐴m 𝐵)) ∧ ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦))))
225fvexi 6669 . . . . . 6 𝐵 ∈ V
234fvexi 6669 . . . . . 6 𝐴 ∈ V
2422, 23elmap 8436 . . . . 5 (𝐹 ∈ (𝐵m 𝐴) ↔ 𝐹:𝐴𝐵)
2523, 22elmap 8436 . . . . 5 (𝐺 ∈ (𝐴m 𝐵) ↔ 𝐺:𝐵𝐴)
2624, 25anbi12i 629 . . . 4 ((𝐹 ∈ (𝐵m 𝐴) ∧ 𝐺 ∈ (𝐴m 𝐵)) ↔ (𝐹:𝐴𝐵𝐺:𝐵𝐴))
2726anbi1i 626 . . 3 (((𝐹 ∈ (𝐵m 𝐴) ∧ 𝐺 ∈ (𝐴m 𝐵)) ∧ ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦))) ↔ ((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦))))
2821, 27bitr2i 279 . 2 (((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦))) ↔ 𝐹{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑔 ∈ (𝐴m 𝐵)) ∧ ∀𝑥𝐴𝑦𝐵 ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦)))}𝐺)
2911, 28bitr4di 292 1 (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106   class class class wbr 5034  {copab 5096  ⟶wf 6328  ‘cfv 6332  (class class class)co 7145   ↑m cmap 8407  Basecbs 16495  lecple 16584   Proset cproset 17548  MGalConncmgc 30731 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 2770  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-opab 5097  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-fv 6340  df-ov 7148  df-oprab 7149  df-mpo 7150  df-map 8409  df-mgc 30733 This theorem is referenced by:  mgcf1  30740  mgcf2  30741  mgccole1  30742  mgccole2  30743  mgcmnt1  30744  mgcmnt2  30745  dfmgc2lem  30747  dfmgc2  30748  mgccnv  30751  pwrssmgc  30752  nsgmgc  31066
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