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Theorem dfmgc2lem 30992
Description: Lemma for dfmgc2, backwards direction. (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 )
dfmgc2lem.1 (𝜑𝐹:𝐴𝐵)
dfmgc2lem.2 (𝜑𝐺:𝐵𝐴)
dfmgc2lem.3 (𝜑 → ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦)))
dfmgc2lem.4 (𝜑 → ∀𝑢𝐵𝑣𝐵 (𝑢 𝑣 → (𝐺𝑢) (𝐺𝑣)))
dfmgc2lem.5 ((𝜑𝑥𝐴) → 𝑥 (𝐺‘(𝐹𝑥)))
dfmgc2lem.6 ((𝜑𝑢𝐵) → (𝐹‘(𝐺𝑢)) 𝑢)
Assertion
Ref Expression
dfmgc2lem (𝜑𝐹𝐻𝐺)
Distinct variable groups:   𝑣,   𝑣,   𝑣,𝐴,𝑥,𝑦   𝑣,𝐵,𝑥,𝑦   𝑣,𝑉,𝑥,𝑦   𝑣,𝑊,𝑥,𝑦   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦   𝑢, ,𝑣   𝑥, ,𝑦   𝑢,   𝑥, ,𝑦   𝑢,𝐴   𝑢,𝐵   𝑢,𝐹,𝑣   𝑢,𝐺,𝑣   𝜑,𝑢   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑦,𝑣)   𝐻(𝑥,𝑦,𝑣,𝑢)   𝑉(𝑢)   𝑊(𝑢)

Proof of Theorem dfmgc2lem
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfmgc2lem.1 . . 3 (𝜑𝐹:𝐴𝐵)
2 dfmgc2lem.2 . . 3 (𝜑𝐺:𝐵𝐴)
31, 2jca 515 . 2 (𝜑 → (𝐹:𝐴𝐵𝐺:𝐵𝐴))
4 mgcval.2 . . . . . . 7 (𝜑𝑉 ∈ Proset )
54ad3antrrr 730 . . . . . 6 ((((𝜑𝑧𝐴) ∧ 𝑤𝐵) ∧ (𝐹𝑧) 𝑤) → 𝑉 ∈ Proset )
6 simplr 769 . . . . . . 7 (((𝜑𝑧𝐴) ∧ 𝑤𝐵) → 𝑧𝐴)
76adantr 484 . . . . . 6 ((((𝜑𝑧𝐴) ∧ 𝑤𝐵) ∧ (𝐹𝑧) 𝑤) → 𝑧𝐴)
82ad3antrrr 730 . . . . . . 7 ((((𝜑𝑧𝐴) ∧ 𝑤𝐵) ∧ (𝐹𝑧) 𝑤) → 𝐺:𝐵𝐴)
91ad3antrrr 730 . . . . . . . 8 ((((𝜑𝑧𝐴) ∧ 𝑤𝐵) ∧ (𝐹𝑧) 𝑤) → 𝐹:𝐴𝐵)
109, 7ffvelrnd 6905 . . . . . . 7 ((((𝜑𝑧𝐴) ∧ 𝑤𝐵) ∧ (𝐹𝑧) 𝑤) → (𝐹𝑧) ∈ 𝐵)
118, 10ffvelrnd 6905 . . . . . 6 ((((𝜑𝑧𝐴) ∧ 𝑤𝐵) ∧ (𝐹𝑧) 𝑤) → (𝐺‘(𝐹𝑧)) ∈ 𝐴)
122ad2antrr 726 . . . . . . . 8 (((𝜑𝑧𝐴) ∧ 𝑤𝐵) → 𝐺:𝐵𝐴)
13 simpr 488 . . . . . . . 8 (((𝜑𝑧𝐴) ∧ 𝑤𝐵) → 𝑤𝐵)
1412, 13ffvelrnd 6905 . . . . . . 7 (((𝜑𝑧𝐴) ∧ 𝑤𝐵) → (𝐺𝑤) ∈ 𝐴)
1514adantr 484 . . . . . 6 ((((𝜑𝑧𝐴) ∧ 𝑤𝐵) ∧ (𝐹𝑧) 𝑤) → (𝐺𝑤) ∈ 𝐴)
16 dfmgc2lem.5 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥 (𝐺‘(𝐹𝑥)))
1716ralrimiva 3105 . . . . . . . 8 (𝜑 → ∀𝑥𝐴 𝑥 (𝐺‘(𝐹𝑥)))
1817ad3antrrr 730 . . . . . . 7 ((((𝜑𝑧𝐴) ∧ 𝑤𝐵) ∧ (𝐹𝑧) 𝑤) → ∀𝑥𝐴 𝑥 (𝐺‘(𝐹𝑥)))
19 simpr 488 . . . . . . . . 9 (((((𝜑𝑧𝐴) ∧ 𝑤𝐵) ∧ (𝐹𝑧) 𝑤) ∧ 𝑥 = 𝑧) → 𝑥 = 𝑧)
2019fveq2d 6721 . . . . . . . . . 10 (((((𝜑𝑧𝐴) ∧ 𝑤𝐵) ∧ (𝐹𝑧) 𝑤) ∧ 𝑥 = 𝑧) → (𝐹𝑥) = (𝐹𝑧))
2120fveq2d 6721 . . . . . . . . 9 (((((𝜑𝑧𝐴) ∧ 𝑤𝐵) ∧ (𝐹𝑧) 𝑤) ∧ 𝑥 = 𝑧) → (𝐺‘(𝐹𝑥)) = (𝐺‘(𝐹𝑧)))
2219, 21breq12d 5066 . . . . . . . 8 (((((𝜑𝑧𝐴) ∧ 𝑤𝐵) ∧ (𝐹𝑧) 𝑤) ∧ 𝑥 = 𝑧) → (𝑥 (𝐺‘(𝐹𝑥)) ↔ 𝑧 (𝐺‘(𝐹𝑧))))
237, 22rspcdv 3529 . . . . . . 7 ((((𝜑𝑧𝐴) ∧ 𝑤𝐵) ∧ (𝐹𝑧) 𝑤) → (∀𝑥𝐴 𝑥 (𝐺‘(𝐹𝑥)) → 𝑧 (𝐺‘(𝐹𝑧))))
2418, 23mpd 15 . . . . . 6 ((((𝜑𝑧𝐴) ∧ 𝑤𝐵) ∧ (𝐹𝑧) 𝑤) → 𝑧 (𝐺‘(𝐹𝑧)))
25 dfmgc2lem.4 . . . . . . . . 9 (𝜑 → ∀𝑢𝐵𝑣𝐵 (𝑢 𝑣 → (𝐺𝑢) (𝐺𝑣)))
2625ad2antrr 726 . . . . . . . 8 (((𝜑𝑧𝐴) ∧ 𝑤𝐵) → ∀𝑢𝐵𝑣𝐵 (𝑢 𝑣 → (𝐺𝑢) (𝐺𝑣)))
27 breq1 5056 . . . . . . . . . 10 (𝑢 = (𝐹𝑧) → (𝑢 𝑣 ↔ (𝐹𝑧) 𝑣))
28 fveq2 6717 . . . . . . . . . . 11 (𝑢 = (𝐹𝑧) → (𝐺𝑢) = (𝐺‘(𝐹𝑧)))
2928breq1d 5063 . . . . . . . . . 10 (𝑢 = (𝐹𝑧) → ((𝐺𝑢) (𝐺𝑣) ↔ (𝐺‘(𝐹𝑧)) (𝐺𝑣)))
3027, 29imbi12d 348 . . . . . . . . 9 (𝑢 = (𝐹𝑧) → ((𝑢 𝑣 → (𝐺𝑢) (𝐺𝑣)) ↔ ((𝐹𝑧) 𝑣 → (𝐺‘(𝐹𝑧)) (𝐺𝑣))))
31 breq2 5057 . . . . . . . . . 10 (𝑣 = 𝑤 → ((𝐹𝑧) 𝑣 ↔ (𝐹𝑧) 𝑤))
32 fveq2 6717 . . . . . . . . . . 11 (𝑣 = 𝑤 → (𝐺𝑣) = (𝐺𝑤))
3332breq2d 5065 . . . . . . . . . 10 (𝑣 = 𝑤 → ((𝐺‘(𝐹𝑧)) (𝐺𝑣) ↔ (𝐺‘(𝐹𝑧)) (𝐺𝑤)))
3431, 33imbi12d 348 . . . . . . . . 9 (𝑣 = 𝑤 → (((𝐹𝑧) 𝑣 → (𝐺‘(𝐹𝑧)) (𝐺𝑣)) ↔ ((𝐹𝑧) 𝑤 → (𝐺‘(𝐹𝑧)) (𝐺𝑤))))
351ffvelrnda 6904 . . . . . . . . . 10 ((𝜑𝑧𝐴) → (𝐹𝑧) ∈ 𝐵)
3635adantr 484 . . . . . . . . 9 (((𝜑𝑧𝐴) ∧ 𝑤𝐵) → (𝐹𝑧) ∈ 𝐵)
37 eqidd 2738 . . . . . . . . 9 ((((𝜑𝑧𝐴) ∧ 𝑤𝐵) ∧ 𝑢 = (𝐹𝑧)) → 𝐵 = 𝐵)
3830, 34, 36, 37, 13rspc2vd 3862 . . . . . . . 8 (((𝜑𝑧𝐴) ∧ 𝑤𝐵) → (∀𝑢𝐵𝑣𝐵 (𝑢 𝑣 → (𝐺𝑢) (𝐺𝑣)) → ((𝐹𝑧) 𝑤 → (𝐺‘(𝐹𝑧)) (𝐺𝑤))))
3926, 38mpd 15 . . . . . . 7 (((𝜑𝑧𝐴) ∧ 𝑤𝐵) → ((𝐹𝑧) 𝑤 → (𝐺‘(𝐹𝑧)) (𝐺𝑤)))
4039imp 410 . . . . . 6 ((((𝜑𝑧𝐴) ∧ 𝑤𝐵) ∧ (𝐹𝑧) 𝑤) → (𝐺‘(𝐹𝑧)) (𝐺𝑤))
41 mgcoval.1 . . . . . . 7 𝐴 = (Base‘𝑉)
42 mgcoval.3 . . . . . . 7 = (le‘𝑉)
4341, 42prstr 17807 . . . . . 6 ((𝑉 ∈ Proset ∧ (𝑧𝐴 ∧ (𝐺‘(𝐹𝑧)) ∈ 𝐴 ∧ (𝐺𝑤) ∈ 𝐴) ∧ (𝑧 (𝐺‘(𝐹𝑧)) ∧ (𝐺‘(𝐹𝑧)) (𝐺𝑤))) → 𝑧 (𝐺𝑤))
445, 7, 11, 15, 24, 40, 43syl132anc 1390 . . . . 5 ((((𝜑𝑧𝐴) ∧ 𝑤𝐵) ∧ (𝐹𝑧) 𝑤) → 𝑧 (𝐺𝑤))
45 mgcval.3 . . . . . . 7 (𝜑𝑊 ∈ Proset )
4645ad3antrrr 730 . . . . . 6 ((((𝜑𝑧𝐴) ∧ 𝑤𝐵) ∧ 𝑧 (𝐺𝑤)) → 𝑊 ∈ Proset )
4735ad2antrr 726 . . . . . 6 ((((𝜑𝑧𝐴) ∧ 𝑤𝐵) ∧ 𝑧 (𝐺𝑤)) → (𝐹𝑧) ∈ 𝐵)
481ad3antrrr 730 . . . . . . 7 ((((𝜑𝑧𝐴) ∧ 𝑤𝐵) ∧ 𝑧 (𝐺𝑤)) → 𝐹:𝐴𝐵)
4914adantr 484 . . . . . . 7 ((((𝜑𝑧𝐴) ∧ 𝑤𝐵) ∧ 𝑧 (𝐺𝑤)) → (𝐺𝑤) ∈ 𝐴)
5048, 49ffvelrnd 6905 . . . . . 6 ((((𝜑𝑧𝐴) ∧ 𝑤𝐵) ∧ 𝑧 (𝐺𝑤)) → (𝐹‘(𝐺𝑤)) ∈ 𝐵)
51 simplr 769 . . . . . 6 ((((𝜑𝑧𝐴) ∧ 𝑤𝐵) ∧ 𝑧 (𝐺𝑤)) → 𝑤𝐵)
52 dfmgc2lem.3 . . . . . . . . 9 (𝜑 → ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦)))
5352ad2antrr 726 . . . . . . . 8 (((𝜑𝑧𝐴) ∧ 𝑤𝐵) → ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦)))
54 breq1 5056 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑥 𝑦𝑧 𝑦))
55 fveq2 6717 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
5655breq1d 5063 . . . . . . . . . 10 (𝑥 = 𝑧 → ((𝐹𝑥) (𝐹𝑦) ↔ (𝐹𝑧) (𝐹𝑦)))
5754, 56imbi12d 348 . . . . . . . . 9 (𝑥 = 𝑧 → ((𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦)) ↔ (𝑧 𝑦 → (𝐹𝑧) (𝐹𝑦))))
58 breq2 5057 . . . . . . . . . 10 (𝑦 = (𝐺𝑤) → (𝑧 𝑦𝑧 (𝐺𝑤)))
59 fveq2 6717 . . . . . . . . . . 11 (𝑦 = (𝐺𝑤) → (𝐹𝑦) = (𝐹‘(𝐺𝑤)))
6059breq2d 5065 . . . . . . . . . 10 (𝑦 = (𝐺𝑤) → ((𝐹𝑧) (𝐹𝑦) ↔ (𝐹𝑧) (𝐹‘(𝐺𝑤))))
6158, 60imbi12d 348 . . . . . . . . 9 (𝑦 = (𝐺𝑤) → ((𝑧 𝑦 → (𝐹𝑧) (𝐹𝑦)) ↔ (𝑧 (𝐺𝑤) → (𝐹𝑧) (𝐹‘(𝐺𝑤)))))
62 eqidd 2738 . . . . . . . . 9 ((((𝜑𝑧𝐴) ∧ 𝑤𝐵) ∧ 𝑥 = 𝑧) → 𝐴 = 𝐴)
6357, 61, 6, 62, 14rspc2vd 3862 . . . . . . . 8 (((𝜑𝑧𝐴) ∧ 𝑤𝐵) → (∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦)) → (𝑧 (𝐺𝑤) → (𝐹𝑧) (𝐹‘(𝐺𝑤)))))
6453, 63mpd 15 . . . . . . 7 (((𝜑𝑧𝐴) ∧ 𝑤𝐵) → (𝑧 (𝐺𝑤) → (𝐹𝑧) (𝐹‘(𝐺𝑤))))
6564imp 410 . . . . . 6 ((((𝜑𝑧𝐴) ∧ 𝑤𝐵) ∧ 𝑧 (𝐺𝑤)) → (𝐹𝑧) (𝐹‘(𝐺𝑤)))
66 dfmgc2lem.6 . . . . . . . . 9 ((𝜑𝑢𝐵) → (𝐹‘(𝐺𝑢)) 𝑢)
6766ralrimiva 3105 . . . . . . . 8 (𝜑 → ∀𝑢𝐵 (𝐹‘(𝐺𝑢)) 𝑢)
6867ad3antrrr 730 . . . . . . 7 ((((𝜑𝑧𝐴) ∧ 𝑤𝐵) ∧ 𝑧 (𝐺𝑤)) → ∀𝑢𝐵 (𝐹‘(𝐺𝑢)) 𝑢)
69 simpr 488 . . . . . . . . . . 11 (((((𝜑𝑧𝐴) ∧ 𝑤𝐵) ∧ 𝑧 (𝐺𝑤)) ∧ 𝑢 = 𝑤) → 𝑢 = 𝑤)
7069fveq2d 6721 . . . . . . . . . 10 (((((𝜑𝑧𝐴) ∧ 𝑤𝐵) ∧ 𝑧 (𝐺𝑤)) ∧ 𝑢 = 𝑤) → (𝐺𝑢) = (𝐺𝑤))
7170fveq2d 6721 . . . . . . . . 9 (((((𝜑𝑧𝐴) ∧ 𝑤𝐵) ∧ 𝑧 (𝐺𝑤)) ∧ 𝑢 = 𝑤) → (𝐹‘(𝐺𝑢)) = (𝐹‘(𝐺𝑤)))
7271, 69breq12d 5066 . . . . . . . 8 (((((𝜑𝑧𝐴) ∧ 𝑤𝐵) ∧ 𝑧 (𝐺𝑤)) ∧ 𝑢 = 𝑤) → ((𝐹‘(𝐺𝑢)) 𝑢 ↔ (𝐹‘(𝐺𝑤)) 𝑤))
7351, 72rspcdv 3529 . . . . . . 7 ((((𝜑𝑧𝐴) ∧ 𝑤𝐵) ∧ 𝑧 (𝐺𝑤)) → (∀𝑢𝐵 (𝐹‘(𝐺𝑢)) 𝑢 → (𝐹‘(𝐺𝑤)) 𝑤))
7468, 73mpd 15 . . . . . 6 ((((𝜑𝑧𝐴) ∧ 𝑤𝐵) ∧ 𝑧 (𝐺𝑤)) → (𝐹‘(𝐺𝑤)) 𝑤)
75 mgcoval.2 . . . . . . 7 𝐵 = (Base‘𝑊)
76 mgcoval.4 . . . . . . 7 = (le‘𝑊)
7775, 76prstr 17807 . . . . . 6 ((𝑊 ∈ Proset ∧ ((𝐹𝑧) ∈ 𝐵 ∧ (𝐹‘(𝐺𝑤)) ∈ 𝐵𝑤𝐵) ∧ ((𝐹𝑧) (𝐹‘(𝐺𝑤)) ∧ (𝐹‘(𝐺𝑤)) 𝑤)) → (𝐹𝑧) 𝑤)
7846, 47, 50, 51, 65, 74, 77syl132anc 1390 . . . . 5 ((((𝜑𝑧𝐴) ∧ 𝑤𝐵) ∧ 𝑧 (𝐺𝑤)) → (𝐹𝑧) 𝑤)
7944, 78impbida 801 . . . 4 (((𝜑𝑧𝐴) ∧ 𝑤𝐵) → ((𝐹𝑧) 𝑤𝑧 (𝐺𝑤)))
8079anasss 470 . . 3 ((𝜑 ∧ (𝑧𝐴𝑤𝐵)) → ((𝐹𝑧) 𝑤𝑧 (𝐺𝑤)))
8180ralrimivva 3112 . 2 (𝜑 → ∀𝑧𝐴𝑤𝐵 ((𝐹𝑧) 𝑤𝑧 (𝐺𝑤)))
82 mgcval.1 . . 3 𝐻 = (𝑉MGalConn𝑊)
8341, 75, 42, 76, 82, 4, 45mgcval 30984 . 2 (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ∀𝑧𝐴𝑤𝐵 ((𝐹𝑧) 𝑤𝑧 (𝐺𝑤)))))
843, 81, 83mpbir2and 713 1 (𝜑𝐹𝐻𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2110  wral 3061   class class class wbr 5053  wf 6376  cfv 6380  (class class class)co 7213  Basecbs 16760  lecple 16809   Proset cproset 17800  MGalConncmgc 30976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-map 8510  df-proset 17802  df-mgc 30978
This theorem is referenced by:  dfmgc2  30993
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