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Theorem mgccole2 32666
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 = (Baseβ€˜π‘‰)
mgcoval.2 𝐡 = (Baseβ€˜π‘Š)
mgcoval.3 ≀ = (leβ€˜π‘‰)
mgcoval.4 ≲ = (leβ€˜π‘Š)
mgcval.1 𝐻 = (𝑉MGalConnπ‘Š)
mgcval.2 (πœ‘ β†’ 𝑉 ∈ Proset )
mgcval.3 (πœ‘ β†’ π‘Š ∈ Proset )
mgccole.1 (πœ‘ β†’ 𝐹𝐻𝐺)
mgccole2.1 (πœ‘ β†’ π‘Œ ∈ 𝐡)
Assertion
Ref Expression
mgccole2 (πœ‘ β†’ (πΉβ€˜(πΊβ€˜π‘Œ)) ≲ π‘Œ)

Proof of Theorem mgccole2
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mgcval.2 . . 3 (πœ‘ β†’ 𝑉 ∈ Proset )
2 mgccole.1 . . . . . 6 (πœ‘ β†’ 𝐹𝐻𝐺)
3 mgcoval.1 . . . . . . 7 𝐴 = (Baseβ€˜π‘‰)
4 mgcoval.2 . . . . . . 7 𝐡 = (Baseβ€˜π‘Š)
5 mgcoval.3 . . . . . . 7 ≀ = (leβ€˜π‘‰)
6 mgcoval.4 . . . . . . 7 ≲ = (leβ€˜π‘Š)
7 mgcval.1 . . . . . . 7 𝐻 = (𝑉MGalConnπ‘Š)
8 mgcval.3 . . . . . . 7 (πœ‘ β†’ π‘Š ∈ Proset )
93, 4, 5, 6, 7, 1, 8mgcval 32662 . . . . . 6 (πœ‘ β†’ (𝐹𝐻𝐺 ↔ ((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐡⟢𝐴) ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (πΊβ€˜π‘¦)))))
102, 9mpbid 231 . . . . 5 (πœ‘ β†’ ((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐡⟢𝐴) ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (πΊβ€˜π‘¦))))
1110simplrd 767 . . . 4 (πœ‘ β†’ 𝐺:𝐡⟢𝐴)
12 mgccole2.1 . . . 4 (πœ‘ β†’ π‘Œ ∈ 𝐡)
1311, 12ffvelcdmd 7081 . . 3 (πœ‘ β†’ (πΊβ€˜π‘Œ) ∈ 𝐴)
143, 5prsref 18264 . . 3 ((𝑉 ∈ Proset ∧ (πΊβ€˜π‘Œ) ∈ 𝐴) β†’ (πΊβ€˜π‘Œ) ≀ (πΊβ€˜π‘Œ))
151, 13, 14syl2anc 583 . 2 (πœ‘ β†’ (πΊβ€˜π‘Œ) ≀ (πΊβ€˜π‘Œ))
1610simprd 495 . . . 4 (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (πΊβ€˜π‘¦)))
17 fveq2 6885 . . . . . . . . 9 (π‘₯ = (πΊβ€˜π‘Œ) β†’ (πΉβ€˜π‘₯) = (πΉβ€˜(πΊβ€˜π‘Œ)))
1817breq1d 5151 . . . . . . . 8 (π‘₯ = (πΊβ€˜π‘Œ) β†’ ((πΉβ€˜π‘₯) ≲ 𝑦 ↔ (πΉβ€˜(πΊβ€˜π‘Œ)) ≲ 𝑦))
19 breq1 5144 . . . . . . . 8 (π‘₯ = (πΊβ€˜π‘Œ) β†’ (π‘₯ ≀ (πΊβ€˜π‘¦) ↔ (πΊβ€˜π‘Œ) ≀ (πΊβ€˜π‘¦)))
2018, 19bibi12d 345 . . . . . . 7 (π‘₯ = (πΊβ€˜π‘Œ) β†’ (((πΉβ€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (πΊβ€˜π‘¦)) ↔ ((πΉβ€˜(πΊβ€˜π‘Œ)) ≲ 𝑦 ↔ (πΊβ€˜π‘Œ) ≀ (πΊβ€˜π‘¦))))
2120adantl 481 . . . . . 6 ((πœ‘ ∧ π‘₯ = (πΊβ€˜π‘Œ)) β†’ (((πΉβ€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (πΊβ€˜π‘¦)) ↔ ((πΉβ€˜(πΊβ€˜π‘Œ)) ≲ 𝑦 ↔ (πΊβ€˜π‘Œ) ≀ (πΊβ€˜π‘¦))))
2221ralbidv 3171 . . . . 5 ((πœ‘ ∧ π‘₯ = (πΊβ€˜π‘Œ)) β†’ (βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (πΊβ€˜π‘¦)) ↔ βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜(πΊβ€˜π‘Œ)) ≲ 𝑦 ↔ (πΊβ€˜π‘Œ) ≀ (πΊβ€˜π‘¦))))
2313, 22rspcdv 3598 . . . 4 (πœ‘ β†’ (βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (πΊβ€˜π‘¦)) β†’ βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜(πΊβ€˜π‘Œ)) ≲ 𝑦 ↔ (πΊβ€˜π‘Œ) ≀ (πΊβ€˜π‘¦))))
2416, 23mpd 15 . . 3 (πœ‘ β†’ βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜(πΊβ€˜π‘Œ)) ≲ 𝑦 ↔ (πΊβ€˜π‘Œ) ≀ (πΊβ€˜π‘¦)))
25 simpr 484 . . . . . 6 ((πœ‘ ∧ 𝑦 = π‘Œ) β†’ 𝑦 = π‘Œ)
2625breq2d 5153 . . . . 5 ((πœ‘ ∧ 𝑦 = π‘Œ) β†’ ((πΉβ€˜(πΊβ€˜π‘Œ)) ≲ 𝑦 ↔ (πΉβ€˜(πΊβ€˜π‘Œ)) ≲ π‘Œ))
27 fveq2 6885 . . . . . . 7 (𝑦 = π‘Œ β†’ (πΊβ€˜π‘¦) = (πΊβ€˜π‘Œ))
2827adantl 481 . . . . . 6 ((πœ‘ ∧ 𝑦 = π‘Œ) β†’ (πΊβ€˜π‘¦) = (πΊβ€˜π‘Œ))
2928breq2d 5153 . . . . 5 ((πœ‘ ∧ 𝑦 = π‘Œ) β†’ ((πΊβ€˜π‘Œ) ≀ (πΊβ€˜π‘¦) ↔ (πΊβ€˜π‘Œ) ≀ (πΊβ€˜π‘Œ)))
3026, 29bibi12d 345 . . . 4 ((πœ‘ ∧ 𝑦 = π‘Œ) β†’ (((πΉβ€˜(πΊβ€˜π‘Œ)) ≲ 𝑦 ↔ (πΊβ€˜π‘Œ) ≀ (πΊβ€˜π‘¦)) ↔ ((πΉβ€˜(πΊβ€˜π‘Œ)) ≲ π‘Œ ↔ (πΊβ€˜π‘Œ) ≀ (πΊβ€˜π‘Œ))))
3112, 30rspcdv 3598 . . 3 (πœ‘ β†’ (βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜(πΊβ€˜π‘Œ)) ≲ 𝑦 ↔ (πΊβ€˜π‘Œ) ≀ (πΊβ€˜π‘¦)) β†’ ((πΉβ€˜(πΊβ€˜π‘Œ)) ≲ π‘Œ ↔ (πΊβ€˜π‘Œ) ≀ (πΊβ€˜π‘Œ))))
3224, 31mpd 15 . 2 (πœ‘ β†’ ((πΉβ€˜(πΊβ€˜π‘Œ)) ≲ π‘Œ ↔ (πΊβ€˜π‘Œ) ≀ (πΊβ€˜π‘Œ)))
3315, 32mpbird 257 1 (πœ‘ β†’ (πΉβ€˜(πΊβ€˜π‘Œ)) ≲ π‘Œ)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055   class class class wbr 5141  βŸΆwf 6533  β€˜cfv 6537  (class class class)co 7405  Basecbs 17153  lecple 17213   Proset cproset 18258  MGalConncmgc 32654
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-map 8824  df-proset 18260  df-mgc 32656
This theorem is referenced by:  mgcmnt2  32668  mgcmntco  32669  dfmgc2  32671  mgcf1olem1  32676  mgcf1olem2  32677
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