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Theorem mgcmnt2 30798
 Description: The upper adjoint 𝐺 of a Galois connection is monotonically increasing. (Contributed by Thierry Arnoux, 26-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 )
mgccole.1 (𝜑𝐹𝐻𝐺)
mgcmnt2.1 (𝜑𝑋𝐵)
mgcmnt2.2 (𝜑𝑌𝐵)
mgcmnt2.3 (𝜑𝑋 𝑌)
Assertion
Ref Expression
mgcmnt2 (𝜑 → (𝐺𝑋) (𝐺𝑌))

Proof of Theorem mgcmnt2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mgcval.3 . . 3 (𝜑𝑊 ∈ Proset )
2 mgccole.1 . . . . . 6 (𝜑𝐹𝐻𝐺)
3 mgcoval.1 . . . . . . 7 𝐴 = (Base‘𝑉)
4 mgcoval.2 . . . . . . 7 𝐵 = (Base‘𝑊)
5 mgcoval.3 . . . . . . 7 = (le‘𝑉)
6 mgcoval.4 . . . . . . 7 = (le‘𝑊)
7 mgcval.1 . . . . . . 7 𝐻 = (𝑉MGalConn𝑊)
8 mgcval.2 . . . . . . 7 (𝜑𝑉 ∈ Proset )
93, 4, 5, 6, 7, 8, 1mgcval 30792 . . . . . 6 (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)))))
102, 9mpbid 235 . . . . 5 (𝜑 → ((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦))))
1110simplld 768 . . . 4 (𝜑𝐹:𝐴𝐵)
1210simplrd 770 . . . . 5 (𝜑𝐺:𝐵𝐴)
13 mgcmnt2.1 . . . . 5 (𝜑𝑋𝐵)
1412, 13ffvelrnd 6844 . . . 4 (𝜑 → (𝐺𝑋) ∈ 𝐴)
1511, 14ffvelrnd 6844 . . 3 (𝜑 → (𝐹‘(𝐺𝑋)) ∈ 𝐵)
16 mgcmnt2.2 . . 3 (𝜑𝑌𝐵)
173, 4, 5, 6, 7, 8, 1, 2, 13mgccole2 30796 . . 3 (𝜑 → (𝐹‘(𝐺𝑋)) 𝑋)
18 mgcmnt2.3 . . 3 (𝜑𝑋 𝑌)
194, 6prstr 17610 . . 3 ((𝑊 ∈ Proset ∧ ((𝐹‘(𝐺𝑋)) ∈ 𝐵𝑋𝐵𝑌𝐵) ∧ ((𝐹‘(𝐺𝑋)) 𝑋𝑋 𝑌)) → (𝐹‘(𝐺𝑋)) 𝑌)
201, 15, 13, 16, 17, 18, 19syl132anc 1386 . 2 (𝜑 → (𝐹‘(𝐺𝑋)) 𝑌)
21 breq2 5037 . . . 4 (𝑦 = 𝑌 → ((𝐹‘(𝐺𝑋)) 𝑦 ↔ (𝐹‘(𝐺𝑋)) 𝑌))
22 fveq2 6659 . . . . 5 (𝑦 = 𝑌 → (𝐺𝑦) = (𝐺𝑌))
2322breq2d 5045 . . . 4 (𝑦 = 𝑌 → ((𝐺𝑋) (𝐺𝑦) ↔ (𝐺𝑋) (𝐺𝑌)))
2421, 23bibi12d 350 . . 3 (𝑦 = 𝑌 → (((𝐹‘(𝐺𝑋)) 𝑦 ↔ (𝐺𝑋) (𝐺𝑦)) ↔ ((𝐹‘(𝐺𝑋)) 𝑌 ↔ (𝐺𝑋) (𝐺𝑌))))
25 fveq2 6659 . . . . . . 7 (𝑥 = (𝐺𝑋) → (𝐹𝑥) = (𝐹‘(𝐺𝑋)))
2625breq1d 5043 . . . . . 6 (𝑥 = (𝐺𝑋) → ((𝐹𝑥) 𝑦 ↔ (𝐹‘(𝐺𝑋)) 𝑦))
27 breq1 5036 . . . . . 6 (𝑥 = (𝐺𝑋) → (𝑥 (𝐺𝑦) ↔ (𝐺𝑋) (𝐺𝑦)))
2826, 27bibi12d 350 . . . . 5 (𝑥 = (𝐺𝑋) → (((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)) ↔ ((𝐹‘(𝐺𝑋)) 𝑦 ↔ (𝐺𝑋) (𝐺𝑦))))
2928ralbidv 3127 . . . 4 (𝑥 = (𝐺𝑋) → (∀𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)) ↔ ∀𝑦𝐵 ((𝐹‘(𝐺𝑋)) 𝑦 ↔ (𝐺𝑋) (𝐺𝑦))))
3010simprd 500 . . . 4 (𝜑 → ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)))
3129, 30, 14rspcdva 3544 . . 3 (𝜑 → ∀𝑦𝐵 ((𝐹‘(𝐺𝑋)) 𝑦 ↔ (𝐺𝑋) (𝐺𝑦)))
3224, 31, 16rspcdva 3544 . 2 (𝜑 → ((𝐹‘(𝐺𝑋)) 𝑌 ↔ (𝐺𝑋) (𝐺𝑌)))
3320, 32mpbid 235 1 (𝜑 → (𝐺𝑋) (𝐺𝑌))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1539   ∈ wcel 2112  ∀wral 3071   class class class wbr 5033  ⟶wf 6332  ‘cfv 6336  (class class class)co 7151  Basecbs 16542  lecple 16631   Proset cproset 17603  MGalConncmgc 30784 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-br 5034  df-opab 5096  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-fv 6344  df-ov 7154  df-oprab 7155  df-mpo 7156  df-map 8419  df-proset 17605  df-mgc 30786 This theorem is referenced by:  dfmgc2  30801  mgcf1olem2  30807
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