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Theorem mgcf1o 30858
Description: Given a Galois connection, exhibit an order isomorphism. (Contributed by Thierry Arnoux, 26-Jul-2024.)
Hypotheses
Ref Expression
mgcf1o.h 𝐻 = (𝑉MGalConn𝑊)
mgcf1o.a 𝐴 = (Base‘𝑉)
mgcf1o.b 𝐵 = (Base‘𝑊)
mgcf1o.1 = (le‘𝑉)
mgcf1o.2 = (le‘𝑊)
mgcf1o.v (𝜑𝑉 ∈ Poset)
mgcf1o.w (𝜑𝑊 ∈ Poset)
mgcf1o.f (𝜑𝐹𝐻𝐺)
Assertion
Ref Expression
mgcf1o (𝜑 → (𝐹 ↾ ran 𝐺) Isom , (ran 𝐺, ran 𝐹))

Proof of Theorem mgcf1o
Dummy variables 𝑢 𝑣 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2738 . . . 4 (𝑥 ∈ ran 𝐺 ↦ (𝐹𝑥)) = (𝑥 ∈ ran 𝐺 ↦ (𝐹𝑥))
2 mgcf1o.f . . . . . . . 8 (𝜑𝐹𝐻𝐺)
3 mgcf1o.a . . . . . . . . 9 𝐴 = (Base‘𝑉)
4 mgcf1o.b . . . . . . . . 9 𝐵 = (Base‘𝑊)
5 mgcf1o.1 . . . . . . . . 9 = (le‘𝑉)
6 mgcf1o.2 . . . . . . . . 9 = (le‘𝑊)
7 mgcf1o.h . . . . . . . . 9 𝐻 = (𝑉MGalConn𝑊)
8 mgcf1o.v . . . . . . . . . 10 (𝜑𝑉 ∈ Poset)
9 posprs 17675 . . . . . . . . . 10 (𝑉 ∈ Poset → 𝑉 ∈ Proset )
108, 9syl 17 . . . . . . . . 9 (𝜑𝑉 ∈ Proset )
11 mgcf1o.w . . . . . . . . . 10 (𝜑𝑊 ∈ Poset)
12 posprs 17675 . . . . . . . . . 10 (𝑊 ∈ Poset → 𝑊 ∈ Proset )
1311, 12syl 17 . . . . . . . . 9 (𝜑𝑊 ∈ Proset )
143, 4, 5, 6, 7, 10, 13dfmgc2 30851 . . . . . . . 8 (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ((∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦)) ∧ ∀𝑢𝐵𝑣𝐵 (𝑢 𝑣 → (𝐺𝑢) (𝐺𝑣))) ∧ (∀𝑢𝐵 (𝐹‘(𝐺𝑢)) 𝑢 ∧ ∀𝑥𝐴 𝑥 (𝐺‘(𝐹𝑥)))))))
152, 14mpbid 235 . . . . . . 7 (𝜑 → ((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ((∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦)) ∧ ∀𝑢𝐵𝑣𝐵 (𝑢 𝑣 → (𝐺𝑢) (𝐺𝑣))) ∧ (∀𝑢𝐵 (𝐹‘(𝐺𝑢)) 𝑢 ∧ ∀𝑥𝐴 𝑥 (𝐺‘(𝐹𝑥))))))
1615simplld 768 . . . . . 6 (𝜑𝐹:𝐴𝐵)
1716ffnd 6505 . . . . 5 (𝜑𝐹 Fn 𝐴)
1815simplrd 770 . . . . . . 7 (𝜑𝐺:𝐵𝐴)
1918frnd 6512 . . . . . 6 (𝜑 → ran 𝐺𝐴)
2019sselda 3877 . . . . 5 ((𝜑𝑥 ∈ ran 𝐺) → 𝑥𝐴)
21 fnfvelrn 6858 . . . . 5 ((𝐹 Fn 𝐴𝑥𝐴) → (𝐹𝑥) ∈ ran 𝐹)
2217, 20, 21syl2an2r 685 . . . 4 ((𝜑𝑥 ∈ ran 𝐺) → (𝐹𝑥) ∈ ran 𝐹)
2318ffnd 6505 . . . . 5 (𝜑𝐺 Fn 𝐵)
2416frnd 6512 . . . . . 6 (𝜑 → ran 𝐹𝐵)
2524sselda 3877 . . . . 5 ((𝜑𝑢 ∈ ran 𝐹) → 𝑢𝐵)
26 fnfvelrn 6858 . . . . 5 ((𝐺 Fn 𝐵𝑢𝐵) → (𝐺𝑢) ∈ ran 𝐺)
2723, 25, 26syl2an2r 685 . . . 4 ((𝜑𝑢 ∈ ran 𝐹) → (𝐺𝑢) ∈ ran 𝐺)
288ad4antr 732 . . . . . . . 8 (((((𝜑 ∧ (𝑥 ∈ ran 𝐺𝑢 ∈ ran 𝐹)) ∧ 𝑥 = (𝐺𝑢)) ∧ 𝑦𝐴) ∧ (𝐹𝑦) = 𝑢) → 𝑉 ∈ Poset)
2911ad4antr 732 . . . . . . . 8 (((((𝜑 ∧ (𝑥 ∈ ran 𝐺𝑢 ∈ ran 𝐹)) ∧ 𝑥 = (𝐺𝑢)) ∧ 𝑦𝐴) ∧ (𝐹𝑦) = 𝑢) → 𝑊 ∈ Poset)
302ad4antr 732 . . . . . . . 8 (((((𝜑 ∧ (𝑥 ∈ ran 𝐺𝑢 ∈ ran 𝐹)) ∧ 𝑥 = (𝐺𝑢)) ∧ 𝑦𝐴) ∧ (𝐹𝑦) = 𝑢) → 𝐹𝐻𝐺)
31 simplr 769 . . . . . . . 8 (((((𝜑 ∧ (𝑥 ∈ ran 𝐺𝑢 ∈ ran 𝐹)) ∧ 𝑥 = (𝐺𝑢)) ∧ 𝑦𝐴) ∧ (𝐹𝑦) = 𝑢) → 𝑦𝐴)
327, 3, 4, 5, 6, 28, 29, 30, 31mgcf1olem1 30856 . . . . . . 7 (((((𝜑 ∧ (𝑥 ∈ ran 𝐺𝑢 ∈ ran 𝐹)) ∧ 𝑥 = (𝐺𝑢)) ∧ 𝑦𝐴) ∧ (𝐹𝑦) = 𝑢) → (𝐹‘(𝐺‘(𝐹𝑦))) = (𝐹𝑦))
33 simpr 488 . . . . . . . . . 10 (((((𝜑 ∧ (𝑥 ∈ ran 𝐺𝑢 ∈ ran 𝐹)) ∧ 𝑥 = (𝐺𝑢)) ∧ 𝑦𝐴) ∧ (𝐹𝑦) = 𝑢) → (𝐹𝑦) = 𝑢)
3433fveq2d 6678 . . . . . . . . 9 (((((𝜑 ∧ (𝑥 ∈ ran 𝐺𝑢 ∈ ran 𝐹)) ∧ 𝑥 = (𝐺𝑢)) ∧ 𝑦𝐴) ∧ (𝐹𝑦) = 𝑢) → (𝐺‘(𝐹𝑦)) = (𝐺𝑢))
35 simpllr 776 . . . . . . . . 9 (((((𝜑 ∧ (𝑥 ∈ ran 𝐺𝑢 ∈ ran 𝐹)) ∧ 𝑥 = (𝐺𝑢)) ∧ 𝑦𝐴) ∧ (𝐹𝑦) = 𝑢) → 𝑥 = (𝐺𝑢))
3634, 35eqtr4d 2776 . . . . . . . 8 (((((𝜑 ∧ (𝑥 ∈ ran 𝐺𝑢 ∈ ran 𝐹)) ∧ 𝑥 = (𝐺𝑢)) ∧ 𝑦𝐴) ∧ (𝐹𝑦) = 𝑢) → (𝐺‘(𝐹𝑦)) = 𝑥)
3736fveq2d 6678 . . . . . . 7 (((((𝜑 ∧ (𝑥 ∈ ran 𝐺𝑢 ∈ ran 𝐹)) ∧ 𝑥 = (𝐺𝑢)) ∧ 𝑦𝐴) ∧ (𝐹𝑦) = 𝑢) → (𝐹‘(𝐺‘(𝐹𝑦))) = (𝐹𝑥))
3832, 37, 333eqtr3rd 2782 . . . . . 6 (((((𝜑 ∧ (𝑥 ∈ ran 𝐺𝑢 ∈ ran 𝐹)) ∧ 𝑥 = (𝐺𝑢)) ∧ 𝑦𝐴) ∧ (𝐹𝑦) = 𝑢) → 𝑢 = (𝐹𝑥))
3917ad2antrr 726 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ ran 𝐺𝑢 ∈ ran 𝐹)) ∧ 𝑥 = (𝐺𝑢)) → 𝐹 Fn 𝐴)
40 simplrr 778 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ ran 𝐺𝑢 ∈ ran 𝐹)) ∧ 𝑥 = (𝐺𝑢)) → 𝑢 ∈ ran 𝐹)
41 fvelrnb 6730 . . . . . . . 8 (𝐹 Fn 𝐴 → (𝑢 ∈ ran 𝐹 ↔ ∃𝑦𝐴 (𝐹𝑦) = 𝑢))
4241biimpa 480 . . . . . . 7 ((𝐹 Fn 𝐴𝑢 ∈ ran 𝐹) → ∃𝑦𝐴 (𝐹𝑦) = 𝑢)
4339, 40, 42syl2anc 587 . . . . . 6 (((𝜑 ∧ (𝑥 ∈ ran 𝐺𝑢 ∈ ran 𝐹)) ∧ 𝑥 = (𝐺𝑢)) → ∃𝑦𝐴 (𝐹𝑦) = 𝑢)
4438, 43r19.29a 3199 . . . . 5 (((𝜑 ∧ (𝑥 ∈ ran 𝐺𝑢 ∈ ran 𝐹)) ∧ 𝑥 = (𝐺𝑢)) → 𝑢 = (𝐹𝑥))
458ad4antr 732 . . . . . . . 8 (((((𝜑 ∧ (𝑥 ∈ ran 𝐺𝑢 ∈ ran 𝐹)) ∧ 𝑢 = (𝐹𝑥)) ∧ 𝑣𝐵) ∧ (𝐺𝑣) = 𝑥) → 𝑉 ∈ Poset)
4611ad4antr 732 . . . . . . . 8 (((((𝜑 ∧ (𝑥 ∈ ran 𝐺𝑢 ∈ ran 𝐹)) ∧ 𝑢 = (𝐹𝑥)) ∧ 𝑣𝐵) ∧ (𝐺𝑣) = 𝑥) → 𝑊 ∈ Poset)
472ad4antr 732 . . . . . . . 8 (((((𝜑 ∧ (𝑥 ∈ ran 𝐺𝑢 ∈ ran 𝐹)) ∧ 𝑢 = (𝐹𝑥)) ∧ 𝑣𝐵) ∧ (𝐺𝑣) = 𝑥) → 𝐹𝐻𝐺)
48 simplr 769 . . . . . . . 8 (((((𝜑 ∧ (𝑥 ∈ ran 𝐺𝑢 ∈ ran 𝐹)) ∧ 𝑢 = (𝐹𝑥)) ∧ 𝑣𝐵) ∧ (𝐺𝑣) = 𝑥) → 𝑣𝐵)
497, 3, 4, 5, 6, 45, 46, 47, 48mgcf1olem2 30857 . . . . . . 7 (((((𝜑 ∧ (𝑥 ∈ ran 𝐺𝑢 ∈ ran 𝐹)) ∧ 𝑢 = (𝐹𝑥)) ∧ 𝑣𝐵) ∧ (𝐺𝑣) = 𝑥) → (𝐺‘(𝐹‘(𝐺𝑣))) = (𝐺𝑣))
50 simpr 488 . . . . . . . . . 10 (((((𝜑 ∧ (𝑥 ∈ ran 𝐺𝑢 ∈ ran 𝐹)) ∧ 𝑢 = (𝐹𝑥)) ∧ 𝑣𝐵) ∧ (𝐺𝑣) = 𝑥) → (𝐺𝑣) = 𝑥)
5150fveq2d 6678 . . . . . . . . 9 (((((𝜑 ∧ (𝑥 ∈ ran 𝐺𝑢 ∈ ran 𝐹)) ∧ 𝑢 = (𝐹𝑥)) ∧ 𝑣𝐵) ∧ (𝐺𝑣) = 𝑥) → (𝐹‘(𝐺𝑣)) = (𝐹𝑥))
52 simpllr 776 . . . . . . . . 9 (((((𝜑 ∧ (𝑥 ∈ ran 𝐺𝑢 ∈ ran 𝐹)) ∧ 𝑢 = (𝐹𝑥)) ∧ 𝑣𝐵) ∧ (𝐺𝑣) = 𝑥) → 𝑢 = (𝐹𝑥))
5351, 52eqtr4d 2776 . . . . . . . 8 (((((𝜑 ∧ (𝑥 ∈ ran 𝐺𝑢 ∈ ran 𝐹)) ∧ 𝑢 = (𝐹𝑥)) ∧ 𝑣𝐵) ∧ (𝐺𝑣) = 𝑥) → (𝐹‘(𝐺𝑣)) = 𝑢)
5453fveq2d 6678 . . . . . . 7 (((((𝜑 ∧ (𝑥 ∈ ran 𝐺𝑢 ∈ ran 𝐹)) ∧ 𝑢 = (𝐹𝑥)) ∧ 𝑣𝐵) ∧ (𝐺𝑣) = 𝑥) → (𝐺‘(𝐹‘(𝐺𝑣))) = (𝐺𝑢))
5549, 54, 503eqtr3rd 2782 . . . . . 6 (((((𝜑 ∧ (𝑥 ∈ ran 𝐺𝑢 ∈ ran 𝐹)) ∧ 𝑢 = (𝐹𝑥)) ∧ 𝑣𝐵) ∧ (𝐺𝑣) = 𝑥) → 𝑥 = (𝐺𝑢))
5623ad2antrr 726 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ ran 𝐺𝑢 ∈ ran 𝐹)) ∧ 𝑢 = (𝐹𝑥)) → 𝐺 Fn 𝐵)
57 simplrl 777 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ ran 𝐺𝑢 ∈ ran 𝐹)) ∧ 𝑢 = (𝐹𝑥)) → 𝑥 ∈ ran 𝐺)
58 fvelrnb 6730 . . . . . . . 8 (𝐺 Fn 𝐵 → (𝑥 ∈ ran 𝐺 ↔ ∃𝑣𝐵 (𝐺𝑣) = 𝑥))
5958biimpa 480 . . . . . . 7 ((𝐺 Fn 𝐵𝑥 ∈ ran 𝐺) → ∃𝑣𝐵 (𝐺𝑣) = 𝑥)
6056, 57, 59syl2anc 587 . . . . . 6 (((𝜑 ∧ (𝑥 ∈ ran 𝐺𝑢 ∈ ran 𝐹)) ∧ 𝑢 = (𝐹𝑥)) → ∃𝑣𝐵 (𝐺𝑣) = 𝑥)
6155, 60r19.29a 3199 . . . . 5 (((𝜑 ∧ (𝑥 ∈ ran 𝐺𝑢 ∈ ran 𝐹)) ∧ 𝑢 = (𝐹𝑥)) → 𝑥 = (𝐺𝑢))
6244, 61impbida 801 . . . 4 ((𝜑 ∧ (𝑥 ∈ ran 𝐺𝑢 ∈ ran 𝐹)) → (𝑥 = (𝐺𝑢) ↔ 𝑢 = (𝐹𝑥)))
631, 22, 27, 62f1o2d 7415 . . 3 (𝜑 → (𝑥 ∈ ran 𝐺 ↦ (𝐹𝑥)):ran 𝐺1-1-onto→ran 𝐹)
6416, 19feqresmpt 6738 . . . 4 (𝜑 → (𝐹 ↾ ran 𝐺) = (𝑥 ∈ ran 𝐺 ↦ (𝐹𝑥)))
6564f1oeq1d 6613 . . 3 (𝜑 → ((𝐹 ↾ ran 𝐺):ran 𝐺1-1-onto→ran 𝐹 ↔ (𝑥 ∈ ran 𝐺 ↦ (𝐹𝑥)):ran 𝐺1-1-onto→ran 𝐹))
6663, 65mpbird 260 . 2 (𝜑 → (𝐹 ↾ ran 𝐺):ran 𝐺1-1-onto→ran 𝐹)
67 simplll 775 . . . . . . 7 ((((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) ∧ 𝑥 𝑦) → 𝜑)
6819ad2antrr 726 . . . . . . . . 9 (((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) → ran 𝐺𝐴)
69 simplr 769 . . . . . . . . 9 (((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) → 𝑥 ∈ ran 𝐺)
7068, 69sseldd 3878 . . . . . . . 8 (((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) → 𝑥𝐴)
7170adantr 484 . . . . . . 7 ((((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) ∧ 𝑥 𝑦) → 𝑥𝐴)
72 simpr 488 . . . . . . . . 9 (((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) → 𝑦 ∈ ran 𝐺)
7368, 72sseldd 3878 . . . . . . . 8 (((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) → 𝑦𝐴)
7473adantr 484 . . . . . . 7 ((((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) ∧ 𝑥 𝑦) → 𝑦𝐴)
75 simpr 488 . . . . . . 7 ((((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) ∧ 𝑥 𝑦) → 𝑥 𝑦)
7615simprld 772 . . . . . . . . . . 11 (𝜑 → (∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦)) ∧ ∀𝑢𝐵𝑣𝐵 (𝑢 𝑣 → (𝐺𝑢) (𝐺𝑣))))
7776simpld 498 . . . . . . . . . 10 (𝜑 → ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦)))
7877r19.21bi 3121 . . . . . . . . 9 ((𝜑𝑥𝐴) → ∀𝑦𝐴 (𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦)))
7978r19.21bi 3121 . . . . . . . 8 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → (𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦)))
8079imp 410 . . . . . . 7 ((((𝜑𝑥𝐴) ∧ 𝑦𝐴) ∧ 𝑥 𝑦) → (𝐹𝑥) (𝐹𝑦))
8167, 71, 74, 75, 80syl1111anc 839 . . . . . 6 ((((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) ∧ 𝑥 𝑦) → (𝐹𝑥) (𝐹𝑦))
8269fvresd 6694 . . . . . . 7 (((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) → ((𝐹 ↾ ran 𝐺)‘𝑥) = (𝐹𝑥))
8382adantr 484 . . . . . 6 ((((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) ∧ 𝑥 𝑦) → ((𝐹 ↾ ran 𝐺)‘𝑥) = (𝐹𝑥))
8472fvresd 6694 . . . . . . 7 (((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) → ((𝐹 ↾ ran 𝐺)‘𝑦) = (𝐹𝑦))
8584adantr 484 . . . . . 6 ((((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) ∧ 𝑥 𝑦) → ((𝐹 ↾ ran 𝐺)‘𝑦) = (𝐹𝑦))
8681, 83, 853brtr4d 5062 . . . . 5 ((((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) ∧ 𝑥 𝑦) → ((𝐹 ↾ ran 𝐺)‘𝑥) ((𝐹 ↾ ran 𝐺)‘𝑦))
8782, 84breq12d 5043 . . . . . . 7 (((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) → (((𝐹 ↾ ran 𝐺)‘𝑥) ((𝐹 ↾ ran 𝐺)‘𝑦) ↔ (𝐹𝑥) (𝐹𝑦)))
8887biimpa 480 . . . . . 6 ((((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) ∧ ((𝐹 ↾ ran 𝐺)‘𝑥) ((𝐹 ↾ ran 𝐺)‘𝑦)) → (𝐹𝑥) (𝐹𝑦))
8911ad7antr 738 . . . . . . . . . . 11 ((((((((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) ∧ (𝐹𝑥) (𝐹𝑦)) ∧ 𝑢𝐵) ∧ (𝐺𝑢) = 𝑥) ∧ 𝑣𝐵) ∧ (𝐺𝑣) = 𝑦) → 𝑊 ∈ Poset)
908ad7antr 738 . . . . . . . . . . 11 ((((((((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) ∧ (𝐹𝑥) (𝐹𝑦)) ∧ 𝑢𝐵) ∧ (𝐺𝑢) = 𝑥) ∧ 𝑣𝐵) ∧ (𝐺𝑣) = 𝑦) → 𝑉 ∈ Poset)
917, 10, 13, 2mgcmnt2d 30853 . . . . . . . . . . . 12 (𝜑𝐺 ∈ (𝑊Monot𝑉))
9291ad7antr 738 . . . . . . . . . . 11 ((((((((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) ∧ (𝐹𝑥) (𝐹𝑦)) ∧ 𝑢𝐵) ∧ (𝐺𝑢) = 𝑥) ∧ 𝑣𝐵) ∧ (𝐺𝑣) = 𝑦) → 𝐺 ∈ (𝑊Monot𝑉))
9316ad7antr 738 . . . . . . . . . . . 12 ((((((((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) ∧ (𝐹𝑥) (𝐹𝑦)) ∧ 𝑢𝐵) ∧ (𝐺𝑢) = 𝑥) ∧ 𝑣𝐵) ∧ (𝐺𝑣) = 𝑦) → 𝐹:𝐴𝐵)
9418ad7antr 738 . . . . . . . . . . . . 13 ((((((((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) ∧ (𝐹𝑥) (𝐹𝑦)) ∧ 𝑢𝐵) ∧ (𝐺𝑢) = 𝑥) ∧ 𝑣𝐵) ∧ (𝐺𝑣) = 𝑦) → 𝐺:𝐵𝐴)
95 simp-4r 784 . . . . . . . . . . . . 13 ((((((((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) ∧ (𝐹𝑥) (𝐹𝑦)) ∧ 𝑢𝐵) ∧ (𝐺𝑢) = 𝑥) ∧ 𝑣𝐵) ∧ (𝐺𝑣) = 𝑦) → 𝑢𝐵)
9694, 95ffvelrnd 6862 . . . . . . . . . . . 12 ((((((((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) ∧ (𝐹𝑥) (𝐹𝑦)) ∧ 𝑢𝐵) ∧ (𝐺𝑢) = 𝑥) ∧ 𝑣𝐵) ∧ (𝐺𝑣) = 𝑦) → (𝐺𝑢) ∈ 𝐴)
9793, 96ffvelrnd 6862 . . . . . . . . . . 11 ((((((((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) ∧ (𝐹𝑥) (𝐹𝑦)) ∧ 𝑢𝐵) ∧ (𝐺𝑢) = 𝑥) ∧ 𝑣𝐵) ∧ (𝐺𝑣) = 𝑦) → (𝐹‘(𝐺𝑢)) ∈ 𝐵)
98 simplr 769 . . . . . . . . . . . . 13 ((((((((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) ∧ (𝐹𝑥) (𝐹𝑦)) ∧ 𝑢𝐵) ∧ (𝐺𝑢) = 𝑥) ∧ 𝑣𝐵) ∧ (𝐺𝑣) = 𝑦) → 𝑣𝐵)
9994, 98ffvelrnd 6862 . . . . . . . . . . . 12 ((((((((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) ∧ (𝐹𝑥) (𝐹𝑦)) ∧ 𝑢𝐵) ∧ (𝐺𝑢) = 𝑥) ∧ 𝑣𝐵) ∧ (𝐺𝑣) = 𝑦) → (𝐺𝑣) ∈ 𝐴)
10093, 99ffvelrnd 6862 . . . . . . . . . . 11 ((((((((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) ∧ (𝐹𝑥) (𝐹𝑦)) ∧ 𝑢𝐵) ∧ (𝐺𝑢) = 𝑥) ∧ 𝑣𝐵) ∧ (𝐺𝑣) = 𝑦) → (𝐹‘(𝐺𝑣)) ∈ 𝐵)
101 simpr 488 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) ∧ (𝐹𝑥) (𝐹𝑦)) → (𝐹𝑥) (𝐹𝑦))
102101ad4antr 732 . . . . . . . . . . . 12 ((((((((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) ∧ (𝐹𝑥) (𝐹𝑦)) ∧ 𝑢𝐵) ∧ (𝐺𝑢) = 𝑥) ∧ 𝑣𝐵) ∧ (𝐺𝑣) = 𝑦) → (𝐹𝑥) (𝐹𝑦))
103 simpllr 776 . . . . . . . . . . . . 13 ((((((((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) ∧ (𝐹𝑥) (𝐹𝑦)) ∧ 𝑢𝐵) ∧ (𝐺𝑢) = 𝑥) ∧ 𝑣𝐵) ∧ (𝐺𝑣) = 𝑦) → (𝐺𝑢) = 𝑥)
104103fveq2d 6678 . . . . . . . . . . . 12 ((((((((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) ∧ (𝐹𝑥) (𝐹𝑦)) ∧ 𝑢𝐵) ∧ (𝐺𝑢) = 𝑥) ∧ 𝑣𝐵) ∧ (𝐺𝑣) = 𝑦) → (𝐹‘(𝐺𝑢)) = (𝐹𝑥))
105 simpr 488 . . . . . . . . . . . . 13 ((((((((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) ∧ (𝐹𝑥) (𝐹𝑦)) ∧ 𝑢𝐵) ∧ (𝐺𝑢) = 𝑥) ∧ 𝑣𝐵) ∧ (𝐺𝑣) = 𝑦) → (𝐺𝑣) = 𝑦)
106105fveq2d 6678 . . . . . . . . . . . 12 ((((((((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) ∧ (𝐹𝑥) (𝐹𝑦)) ∧ 𝑢𝐵) ∧ (𝐺𝑢) = 𝑥) ∧ 𝑣𝐵) ∧ (𝐺𝑣) = 𝑦) → (𝐹‘(𝐺𝑣)) = (𝐹𝑦))
107102, 104, 1063brtr4d 5062 . . . . . . . . . . 11 ((((((((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) ∧ (𝐹𝑥) (𝐹𝑦)) ∧ 𝑢𝐵) ∧ (𝐺𝑢) = 𝑥) ∧ 𝑣𝐵) ∧ (𝐺𝑣) = 𝑦) → (𝐹‘(𝐺𝑢)) (𝐹‘(𝐺𝑣)))
1084, 3, 6, 5, 89, 90, 92, 97, 100, 107ismntd 30839 . . . . . . . . . 10 ((((((((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) ∧ (𝐹𝑥) (𝐹𝑦)) ∧ 𝑢𝐵) ∧ (𝐺𝑢) = 𝑥) ∧ 𝑣𝐵) ∧ (𝐺𝑣) = 𝑦) → (𝐺‘(𝐹‘(𝐺𝑢))) (𝐺‘(𝐹‘(𝐺𝑣))))
1092ad7antr 738 . . . . . . . . . . 11 ((((((((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) ∧ (𝐹𝑥) (𝐹𝑦)) ∧ 𝑢𝐵) ∧ (𝐺𝑢) = 𝑥) ∧ 𝑣𝐵) ∧ (𝐺𝑣) = 𝑦) → 𝐹𝐻𝐺)
1107, 3, 4, 5, 6, 90, 89, 109, 95mgcf1olem2 30857 . . . . . . . . . 10 ((((((((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) ∧ (𝐹𝑥) (𝐹𝑦)) ∧ 𝑢𝐵) ∧ (𝐺𝑢) = 𝑥) ∧ 𝑣𝐵) ∧ (𝐺𝑣) = 𝑦) → (𝐺‘(𝐹‘(𝐺𝑢))) = (𝐺𝑢))
1117, 3, 4, 5, 6, 90, 89, 109, 98mgcf1olem2 30857 . . . . . . . . . 10 ((((((((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) ∧ (𝐹𝑥) (𝐹𝑦)) ∧ 𝑢𝐵) ∧ (𝐺𝑢) = 𝑥) ∧ 𝑣𝐵) ∧ (𝐺𝑣) = 𝑦) → (𝐺‘(𝐹‘(𝐺𝑣))) = (𝐺𝑣))
112108, 110, 1113brtr3d 5061 . . . . . . . . 9 ((((((((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) ∧ (𝐹𝑥) (𝐹𝑦)) ∧ 𝑢𝐵) ∧ (𝐺𝑢) = 𝑥) ∧ 𝑣𝐵) ∧ (𝐺𝑣) = 𝑦) → (𝐺𝑢) (𝐺𝑣))
113112, 103, 1053brtr3d 5061 . . . . . . . 8 ((((((((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) ∧ (𝐹𝑥) (𝐹𝑦)) ∧ 𝑢𝐵) ∧ (𝐺𝑢) = 𝑥) ∧ 𝑣𝐵) ∧ (𝐺𝑣) = 𝑦) → 𝑥 𝑦)
11423ad3antrrr 730 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) ∧ (𝐹𝑥) (𝐹𝑦)) → 𝐺 Fn 𝐵)
115114ad2antrr 726 . . . . . . . . 9 ((((((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) ∧ (𝐹𝑥) (𝐹𝑦)) ∧ 𝑢𝐵) ∧ (𝐺𝑢) = 𝑥) → 𝐺 Fn 𝐵)
116 simp-4r 784 . . . . . . . . 9 ((((((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) ∧ (𝐹𝑥) (𝐹𝑦)) ∧ 𝑢𝐵) ∧ (𝐺𝑢) = 𝑥) → 𝑦 ∈ ran 𝐺)
117 fvelrnb 6730 . . . . . . . . . 10 (𝐺 Fn 𝐵 → (𝑦 ∈ ran 𝐺 ↔ ∃𝑣𝐵 (𝐺𝑣) = 𝑦))
118117biimpa 480 . . . . . . . . 9 ((𝐺 Fn 𝐵𝑦 ∈ ran 𝐺) → ∃𝑣𝐵 (𝐺𝑣) = 𝑦)
119115, 116, 118syl2anc 587 . . . . . . . 8 ((((((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) ∧ (𝐹𝑥) (𝐹𝑦)) ∧ 𝑢𝐵) ∧ (𝐺𝑢) = 𝑥) → ∃𝑣𝐵 (𝐺𝑣) = 𝑦)
120113, 119r19.29a 3199 . . . . . . 7 ((((((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) ∧ (𝐹𝑥) (𝐹𝑦)) ∧ 𝑢𝐵) ∧ (𝐺𝑢) = 𝑥) → 𝑥 𝑦)
121 simpllr 776 . . . . . . . 8 ((((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) ∧ (𝐹𝑥) (𝐹𝑦)) → 𝑥 ∈ ran 𝐺)
122 fvelrnb 6730 . . . . . . . . 9 (𝐺 Fn 𝐵 → (𝑥 ∈ ran 𝐺 ↔ ∃𝑢𝐵 (𝐺𝑢) = 𝑥))
123122biimpa 480 . . . . . . . 8 ((𝐺 Fn 𝐵𝑥 ∈ ran 𝐺) → ∃𝑢𝐵 (𝐺𝑢) = 𝑥)
124114, 121, 123syl2anc 587 . . . . . . 7 ((((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) ∧ (𝐹𝑥) (𝐹𝑦)) → ∃𝑢𝐵 (𝐺𝑢) = 𝑥)
125120, 124r19.29a 3199 . . . . . 6 ((((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) ∧ (𝐹𝑥) (𝐹𝑦)) → 𝑥 𝑦)
12688, 125syldan 594 . . . . 5 ((((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) ∧ ((𝐹 ↾ ran 𝐺)‘𝑥) ((𝐹 ↾ ran 𝐺)‘𝑦)) → 𝑥 𝑦)
12786, 126impbida 801 . . . 4 (((𝜑𝑥 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐺) → (𝑥 𝑦 ↔ ((𝐹 ↾ ran 𝐺)‘𝑥) ((𝐹 ↾ ran 𝐺)‘𝑦)))
128127anasss 470 . . 3 ((𝜑 ∧ (𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺)) → (𝑥 𝑦 ↔ ((𝐹 ↾ ran 𝐺)‘𝑥) ((𝐹 ↾ ran 𝐺)‘𝑦)))
129128ralrimivva 3103 . 2 (𝜑 → ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺(𝑥 𝑦 ↔ ((𝐹 ↾ ran 𝐺)‘𝑥) ((𝐹 ↾ ran 𝐺)‘𝑦)))
130 df-isom 6348 . 2 ((𝐹 ↾ ran 𝐺) Isom , (ran 𝐺, ran 𝐹) ↔ ((𝐹 ↾ ran 𝐺):ran 𝐺1-1-onto→ran 𝐹 ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺(𝑥 𝑦 ↔ ((𝐹 ↾ ran 𝐺)‘𝑥) ((𝐹 ↾ ran 𝐺)‘𝑦))))
13166, 129, 130sylanbrc 586 1 (𝜑 → (𝐹 ↾ ran 𝐺) Isom , (ran 𝐺, ran 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  wcel 2114  wral 3053  wrex 3054  wss 3843   class class class wbr 5030  cmpt 5110  ran crn 5526  cres 5527   Fn wfn 6334  wf 6335  1-1-ontowf1o 6338  cfv 6339   Isom wiso 6340  (class class class)co 7170  Basecbs 16586  lecple 16675   Proset cproset 17652  Posetcpo 17666  Monotcmnt 30833  MGalConncmgc 30834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-isom 6348  df-ov 7173  df-oprab 7174  df-mpo 7175  df-map 8439  df-proset 17654  df-poset 17672  df-mnt 30835  df-mgc 30836
This theorem is referenced by:  nsgqusf1o  31173
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