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Theorem mgccole1 31675
Description: An inequality for the kernel operator 𝐺𝐹. (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)
mgccole1.2 (𝜑𝑋𝐴)
Assertion
Ref Expression
mgccole1 (𝜑𝑋 (𝐺‘(𝐹𝑋)))

Proof of Theorem mgccole1
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 31672 . . . . . 6 (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)))))
102, 9mpbid 231 . . . . 5 (𝜑 → ((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦))))
1110simplld 767 . . . 4 (𝜑𝐹:𝐴𝐵)
12 mgccole1.2 . . . 4 (𝜑𝑋𝐴)
1311, 12ffvelcdmd 7033 . . 3 (𝜑 → (𝐹𝑋) ∈ 𝐵)
144, 6prsref 18148 . . 3 ((𝑊 ∈ Proset ∧ (𝐹𝑋) ∈ 𝐵) → (𝐹𝑋) (𝐹𝑋))
151, 13, 14syl2anc 585 . 2 (𝜑 → (𝐹𝑋) (𝐹𝑋))
16 fveq2 6840 . . . . . . 7 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
1716breq1d 5114 . . . . . 6 (𝑥 = 𝑋 → ((𝐹𝑥) 𝑦 ↔ (𝐹𝑋) 𝑦))
18 breq1 5107 . . . . . 6 (𝑥 = 𝑋 → (𝑥 (𝐺𝑦) ↔ 𝑋 (𝐺𝑦)))
1917, 18bibi12d 346 . . . . 5 (𝑥 = 𝑋 → (((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)) ↔ ((𝐹𝑋) 𝑦𝑋 (𝐺𝑦))))
2019ralbidv 3173 . . . 4 (𝑥 = 𝑋 → (∀𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)) ↔ ∀𝑦𝐵 ((𝐹𝑋) 𝑦𝑋 (𝐺𝑦))))
2110simprd 497 . . . 4 (𝜑 → ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)))
2220, 21, 12rspcdva 3581 . . 3 (𝜑 → ∀𝑦𝐵 ((𝐹𝑋) 𝑦𝑋 (𝐺𝑦)))
23 simpr 486 . . . . . 6 ((𝜑𝑦 = (𝐹𝑋)) → 𝑦 = (𝐹𝑋))
2423breq2d 5116 . . . . 5 ((𝜑𝑦 = (𝐹𝑋)) → ((𝐹𝑋) 𝑦 ↔ (𝐹𝑋) (𝐹𝑋)))
2523fveq2d 6844 . . . . . 6 ((𝜑𝑦 = (𝐹𝑋)) → (𝐺𝑦) = (𝐺‘(𝐹𝑋)))
2625breq2d 5116 . . . . 5 ((𝜑𝑦 = (𝐹𝑋)) → (𝑋 (𝐺𝑦) ↔ 𝑋 (𝐺‘(𝐹𝑋))))
2724, 26bibi12d 346 . . . 4 ((𝜑𝑦 = (𝐹𝑋)) → (((𝐹𝑋) 𝑦𝑋 (𝐺𝑦)) ↔ ((𝐹𝑋) (𝐹𝑋) ↔ 𝑋 (𝐺‘(𝐹𝑋)))))
2813, 27rspcdv 3572 . . 3 (𝜑 → (∀𝑦𝐵 ((𝐹𝑋) 𝑦𝑋 (𝐺𝑦)) → ((𝐹𝑋) (𝐹𝑋) ↔ 𝑋 (𝐺‘(𝐹𝑋)))))
2922, 28mpd 15 . 2 (𝜑 → ((𝐹𝑋) (𝐹𝑋) ↔ 𝑋 (𝐺‘(𝐹𝑋))))
3015, 29mpbid 231 1 (𝜑𝑋 (𝐺‘(𝐹𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3063   class class class wbr 5104  wf 6490  cfv 6494  (class class class)co 7352  Basecbs 17043  lecple 17100   Proset cproset 18142  MGalConncmgc 31664
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7665
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-id 5530  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-iota 6446  df-fun 6496  df-fn 6497  df-f 6498  df-fv 6502  df-ov 7355  df-oprab 7356  df-mpo 7357  df-map 8726  df-proset 18144  df-mgc 31666
This theorem is referenced by:  mgcmnt1  31677  mgcmntco  31679  dfmgc2  31681  mgcf1olem1  31686  mgcf1olem2  31687
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