Theorem mgccole2 33545

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 33541	. . . . . 6 ⊢ (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)))))
10	2, 9	mpbid 235	. . . . 5 ⊢ (𝜑 → ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦))))
11	10	simplrd 782	. . . 4 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
12		mgccole2.1	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
13	11, 12	ffvelcdmd 7083	. . 3 ⊢ (𝜑 → (𝐺‘𝑌) ∈ 𝐴)
14	3, 5	prsref 18465	. . 3 ⊢ ((𝑉 ∈ Proset ∧ (𝐺‘𝑌) ∈ 𝐴) → (𝐺‘𝑌) ≤ (𝐺‘𝑌))
15	1, 13, 14	syl2anc 596	. 2 ⊢ (𝜑 → (𝐺‘𝑌) ≤ (𝐺‘𝑌))
16	10	simprd 501	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)))
17		fveq2 6883	. . . . . . . . 9 ⊢ (𝑥 = (𝐺‘𝑌) → (𝐹‘𝑥) = (𝐹‘(𝐺‘𝑌)))
18	17	breq1d 5113	. . . . . . . 8 ⊢ (𝑥 = (𝐺‘𝑌) → ((𝐹‘𝑥) ≲ 𝑦 ↔ (𝐹‘(𝐺‘𝑌)) ≲ 𝑦))
19		breq1 5106	. . . . . . . 8 ⊢ (𝑥 = (𝐺‘𝑌) → (𝑥 ≤ (𝐺‘𝑦) ↔ (𝐺‘𝑌) ≤ (𝐺‘𝑦)))
20	18, 19	bibi12d 348	. . . . . . 7 ⊢ (𝑥 = (𝐺‘𝑌) → (((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)) ↔ ((𝐹‘(𝐺‘𝑌)) ≲ 𝑦 ↔ (𝐺‘𝑌) ≤ (𝐺‘𝑦))))
21	20	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = (𝐺‘𝑌)) → (((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)) ↔ ((𝐹‘(𝐺‘𝑌)) ≲ 𝑦 ↔ (𝐺‘𝑌) ≤ (𝐺‘𝑦))))
22	21	ralbidv 3186	. . . . 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 3569	. . . 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 16	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ((𝐹‘(𝐺‘𝑌)) ≲ 𝑦 ↔ (𝐺‘𝑌) ≤ (𝐺‘𝑦)))
25		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌)
26	25	breq2d 5115	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝑌) → ((𝐹‘(𝐺‘𝑌)) ≲ 𝑦 ↔ (𝐹‘(𝐺‘𝑌)) ≲ 𝑌))
27		fveq2 6883	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝐺‘𝑦) = (𝐺‘𝑌))
28	27	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝑌) → (𝐺‘𝑦) = (𝐺‘𝑌))
29	28	breq2d 5115	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝑌) → ((𝐺‘𝑌) ≤ (𝐺‘𝑦) ↔ (𝐺‘𝑌) ≤ (𝐺‘𝑌)))
30	26, 29	bibi12d 348	. . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝑌) → (((𝐹‘(𝐺‘𝑌)) ≲ 𝑦 ↔ (𝐺‘𝑌) ≤ (𝐺‘𝑦)) ↔ ((𝐹‘(𝐺‘𝑌)) ≲ 𝑌 ↔ (𝐺‘𝑌) ≤ (𝐺‘𝑌))))
31	12, 30	rspcdv 3569	. . 3 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 ((𝐹‘(𝐺‘𝑌)) ≲ 𝑦 ↔ (𝐺‘𝑌) ≤ (𝐺‘𝑦)) → ((𝐹‘(𝐺‘𝑌)) ≲ 𝑌 ↔ (𝐺‘𝑌) ≤ (𝐺‘𝑌))))
32	24, 31	mpd 16	. 2 ⊢ (𝜑 → ((𝐹‘(𝐺‘𝑌)) ≲ 𝑌 ↔ (𝐺‘𝑌) ≤ (𝐺‘𝑌)))
33	15, 32	mpbird 260	1 ⊢ (𝜑 → (𝐹‘(𝐺‘𝑌)) ≲ 𝑌)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077   class class class wbr 5103  ⟶wf 6533  ‘cfv 6537  (class class class)co 7418  Basecbs 17380  lecple 17428   Proset cproset 18459  MGalConncmgc 33533

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 7749

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 3902  df-un 3904  df-in 3906  df-ss 3916  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 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7421  df-oprab 7422  df-mpo 7423  df-map 8842  df-proset 18461  df-mgc 33535

This theorem is used by:  mgcmnt2  33547  mgcmntco  33548  dfmgc2  33550  mgcf1olem1  33555  mgcf1olem2  33556