Theorem mgccole2 32964

Description: Inequality for the closure operator (𝐹 ∘ 𝐺) of the Galois connection 𝐻. (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	⊢ (𝜑 → 𝑉 ∈ Proset )
mgcval.3	⊢ (𝜑 → 𝑊 ∈ Proset )
mgccole.1	⊢ (𝜑 → 𝐹𝐻𝐺)
mgccole2.1	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
mgccole2	⊢ (𝜑 → (𝐹‘(𝐺‘𝑌)) ≲ 𝑌)

Proof of Theorem mgccole2

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mgcval.2	. . 3 ⊢ (𝜑 → 𝑉 ∈ Proset )
2		mgccole.1	. . . . . 6 ⊢ (𝜑 → 𝐹𝐻𝐺)
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.3	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Proset )
9	3, 4, 5, 6, 7, 1, 8	mgcval 32960	. . . . . 6 ⊢ (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)))))
10	2, 9	mpbid 232	. . . . 5 ⊢ (𝜑 → ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦))))
11	10	simplrd 769	. . . 4 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
12		mgccole2.1	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
13	11, 12	ffvelcdmd 7013	. . 3 ⊢ (𝜑 → (𝐺‘𝑌) ∈ 𝐴)
14	3, 5	prsref 18199	. . 3 ⊢ ((𝑉 ∈ Proset ∧ (𝐺‘𝑌) ∈ 𝐴) → (𝐺‘𝑌) ≤ (𝐺‘𝑌))
15	1, 13, 14	syl2anc 584	. 2 ⊢ (𝜑 → (𝐺‘𝑌) ≤ (𝐺‘𝑌))
16	10	simprd 495	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)))
17		fveq2 6817	. . . . . . . . 9 ⊢ (𝑥 = (𝐺‘𝑌) → (𝐹‘𝑥) = (𝐹‘(𝐺‘𝑌)))
18	17	breq1d 5096	. . . . . . . 8 ⊢ (𝑥 = (𝐺‘𝑌) → ((𝐹‘𝑥) ≲ 𝑦 ↔ (𝐹‘(𝐺‘𝑌)) ≲ 𝑦))
19		breq1 5089	. . . . . . . 8 ⊢ (𝑥 = (𝐺‘𝑌) → (𝑥 ≤ (𝐺‘𝑦) ↔ (𝐺‘𝑌) ≤ (𝐺‘𝑦)))
20	18, 19	bibi12d 345	. . . . . . 7 ⊢ (𝑥 = (𝐺‘𝑌) → (((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)) ↔ ((𝐹‘(𝐺‘𝑌)) ≲ 𝑦 ↔ (𝐺‘𝑌) ≤ (𝐺‘𝑦))))
21	20	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = (𝐺‘𝑌)) → (((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)) ↔ ((𝐹‘(𝐺‘𝑌)) ≲ 𝑦 ↔ (𝐺‘𝑌) ≤ (𝐺‘𝑦))))
22	21	ralbidv 3155	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = (𝐺‘𝑌)) → (∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)) ↔ ∀𝑦 ∈ 𝐵 ((𝐹‘(𝐺‘𝑌)) ≲ 𝑦 ↔ (𝐺‘𝑌) ≤ (𝐺‘𝑦))))
23	13, 22	rspcdv 3564	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)) → ∀𝑦 ∈ 𝐵 ((𝐹‘(𝐺‘𝑌)) ≲ 𝑦 ↔ (𝐺‘𝑌) ≤ (𝐺‘𝑦))))
24	16, 23	mpd 15	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ((𝐹‘(𝐺‘𝑌)) ≲ 𝑦 ↔ (𝐺‘𝑌) ≤ (𝐺‘𝑦)))
25		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌)
26	25	breq2d 5098	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝑌) → ((𝐹‘(𝐺‘𝑌)) ≲ 𝑦 ↔ (𝐹‘(𝐺‘𝑌)) ≲ 𝑌))
27		fveq2 6817	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝐺‘𝑦) = (𝐺‘𝑌))
28	27	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝑌) → (𝐺‘𝑦) = (𝐺‘𝑌))
29	28	breq2d 5098	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝑌) → ((𝐺‘𝑌) ≤ (𝐺‘𝑦) ↔ (𝐺‘𝑌) ≤ (𝐺‘𝑌)))
30	26, 29	bibi12d 345	. . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝑌) → (((𝐹‘(𝐺‘𝑌)) ≲ 𝑦 ↔ (𝐺‘𝑌) ≤ (𝐺‘𝑦)) ↔ ((𝐹‘(𝐺‘𝑌)) ≲ 𝑌 ↔ (𝐺‘𝑌) ≤ (𝐺‘𝑌))))
31	12, 30	rspcdv 3564	. . 3 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 ((𝐹‘(𝐺‘𝑌)) ≲ 𝑦 ↔ (𝐺‘𝑌) ≤ (𝐺‘𝑦)) → ((𝐹‘(𝐺‘𝑌)) ≲ 𝑌 ↔ (𝐺‘𝑌) ≤ (𝐺‘𝑌))))
32	24, 31	mpd 15	. 2 ⊢ (𝜑 → ((𝐹‘(𝐺‘𝑌)) ≲ 𝑌 ↔ (𝐺‘𝑌) ≤ (𝐺‘𝑌)))
33	15, 32	mpbird 257	1 ⊢ (𝜑 → (𝐹‘(𝐺‘𝑌)) ≲ 𝑌)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047   class class class wbr 5086  ⟶wf 6472  ‘cfv 6476  (class class class)co 7341  Basecbs 17115  lecple 17163   Proset cproset 18193  MGalConncmgc 32952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-fv 6484  df-ov 7344  df-oprab 7345  df-mpo 7346  df-map 8747  df-proset 18195  df-mgc 32954

This theorem is referenced by:  mgcmnt2  32966  mgcmntco  32967  dfmgc2  32969  mgcf1olem1  32974  mgcf1olem2  32975