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Theorem soisoi 6770
Description: Infer isomorphism from one direction of an order proof for isomorphisms between strict orders. (Contributed by Stefan O'Rear, 2-Nov-2014.)
Assertion
Ref Expression
soisoi (((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
Distinct variable groups:   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦   𝑥,𝐻,𝑦   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦

Proof of Theorem soisoi
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simprl 787 . . . . 5 (((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) → 𝐻:𝐴onto𝐵)
2 fof 6298 . . . . 5 (𝐻:𝐴onto𝐵𝐻:𝐴𝐵)
31, 2syl 17 . . . 4 (((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) → 𝐻:𝐴𝐵)
4 simpll 783 . . . . . . . 8 (((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) → 𝑅 Or 𝐴)
5 sotrieq 5225 . . . . . . . . 9 ((𝑅 Or 𝐴 ∧ (𝑎𝐴𝑏𝐴)) → (𝑎 = 𝑏 ↔ ¬ (𝑎𝑅𝑏𝑏𝑅𝑎)))
65con2bid 345 . . . . . . . 8 ((𝑅 Or 𝐴 ∧ (𝑎𝐴𝑏𝐴)) → ((𝑎𝑅𝑏𝑏𝑅𝑎) ↔ ¬ 𝑎 = 𝑏))
74, 6sylan 575 . . . . . . 7 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → ((𝑎𝑅𝑏𝑏𝑅𝑎) ↔ ¬ 𝑎 = 𝑏))
8 simprr 789 . . . . . . . . . 10 (((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) → ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))
9 breq1 4812 . . . . . . . . . . . . 13 (𝑥 = 𝑎 → (𝑥𝑅𝑦𝑎𝑅𝑦))
10 fveq2 6375 . . . . . . . . . . . . . 14 (𝑥 = 𝑎 → (𝐻𝑥) = (𝐻𝑎))
1110breq1d 4819 . . . . . . . . . . . . 13 (𝑥 = 𝑎 → ((𝐻𝑥)𝑆(𝐻𝑦) ↔ (𝐻𝑎)𝑆(𝐻𝑦)))
129, 11imbi12d 335 . . . . . . . . . . . 12 (𝑥 = 𝑎 → ((𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)) ↔ (𝑎𝑅𝑦 → (𝐻𝑎)𝑆(𝐻𝑦))))
13 breq2 4813 . . . . . . . . . . . . 13 (𝑦 = 𝑏 → (𝑎𝑅𝑦𝑎𝑅𝑏))
14 fveq2 6375 . . . . . . . . . . . . . 14 (𝑦 = 𝑏 → (𝐻𝑦) = (𝐻𝑏))
1514breq2d 4821 . . . . . . . . . . . . 13 (𝑦 = 𝑏 → ((𝐻𝑎)𝑆(𝐻𝑦) ↔ (𝐻𝑎)𝑆(𝐻𝑏)))
1613, 15imbi12d 335 . . . . . . . . . . . 12 (𝑦 = 𝑏 → ((𝑎𝑅𝑦 → (𝐻𝑎)𝑆(𝐻𝑦)) ↔ (𝑎𝑅𝑏 → (𝐻𝑎)𝑆(𝐻𝑏))))
1712, 16rspc2va 3475 . . . . . . . . . . 11 (((𝑎𝐴𝑏𝐴) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦))) → (𝑎𝑅𝑏 → (𝐻𝑎)𝑆(𝐻𝑏)))
1817ancoms 450 . . . . . . . . . 10 ((∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)) ∧ (𝑎𝐴𝑏𝐴)) → (𝑎𝑅𝑏 → (𝐻𝑎)𝑆(𝐻𝑏)))
198, 18sylan 575 . . . . . . . . 9 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → (𝑎𝑅𝑏 → (𝐻𝑎)𝑆(𝐻𝑏)))
20 simpllr 793 . . . . . . . . . . 11 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → 𝑆 Po 𝐵)
21 simplrl 795 . . . . . . . . . . . . 13 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → 𝐻:𝐴onto𝐵)
2221, 2syl 17 . . . . . . . . . . . 12 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → 𝐻:𝐴𝐵)
23 simprr 789 . . . . . . . . . . . 12 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → 𝑏𝐴)
2422, 23ffvelrnd 6550 . . . . . . . . . . 11 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → (𝐻𝑏) ∈ 𝐵)
25 poirr 5209 . . . . . . . . . . . 12 ((𝑆 Po 𝐵 ∧ (𝐻𝑏) ∈ 𝐵) → ¬ (𝐻𝑏)𝑆(𝐻𝑏))
26 breq1 4812 . . . . . . . . . . . . 13 ((𝐻𝑎) = (𝐻𝑏) → ((𝐻𝑎)𝑆(𝐻𝑏) ↔ (𝐻𝑏)𝑆(𝐻𝑏)))
2726notbid 309 . . . . . . . . . . . 12 ((𝐻𝑎) = (𝐻𝑏) → (¬ (𝐻𝑎)𝑆(𝐻𝑏) ↔ ¬ (𝐻𝑏)𝑆(𝐻𝑏)))
2825, 27syl5ibrcom 238 . . . . . . . . . . 11 ((𝑆 Po 𝐵 ∧ (𝐻𝑏) ∈ 𝐵) → ((𝐻𝑎) = (𝐻𝑏) → ¬ (𝐻𝑎)𝑆(𝐻𝑏)))
2920, 24, 28syl2anc 579 . . . . . . . . . 10 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → ((𝐻𝑎) = (𝐻𝑏) → ¬ (𝐻𝑎)𝑆(𝐻𝑏)))
3029con2d 131 . . . . . . . . 9 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → ((𝐻𝑎)𝑆(𝐻𝑏) → ¬ (𝐻𝑎) = (𝐻𝑏)))
3119, 30syld 47 . . . . . . . 8 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → (𝑎𝑅𝑏 → ¬ (𝐻𝑎) = (𝐻𝑏)))
32 breq1 4812 . . . . . . . . . . . . . 14 (𝑥 = 𝑏 → (𝑥𝑅𝑦𝑏𝑅𝑦))
33 fveq2 6375 . . . . . . . . . . . . . . 15 (𝑥 = 𝑏 → (𝐻𝑥) = (𝐻𝑏))
3433breq1d 4819 . . . . . . . . . . . . . 14 (𝑥 = 𝑏 → ((𝐻𝑥)𝑆(𝐻𝑦) ↔ (𝐻𝑏)𝑆(𝐻𝑦)))
3532, 34imbi12d 335 . . . . . . . . . . . . 13 (𝑥 = 𝑏 → ((𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)) ↔ (𝑏𝑅𝑦 → (𝐻𝑏)𝑆(𝐻𝑦))))
36 breq2 4813 . . . . . . . . . . . . . 14 (𝑦 = 𝑎 → (𝑏𝑅𝑦𝑏𝑅𝑎))
37 fveq2 6375 . . . . . . . . . . . . . . 15 (𝑦 = 𝑎 → (𝐻𝑦) = (𝐻𝑎))
3837breq2d 4821 . . . . . . . . . . . . . 14 (𝑦 = 𝑎 → ((𝐻𝑏)𝑆(𝐻𝑦) ↔ (𝐻𝑏)𝑆(𝐻𝑎)))
3936, 38imbi12d 335 . . . . . . . . . . . . 13 (𝑦 = 𝑎 → ((𝑏𝑅𝑦 → (𝐻𝑏)𝑆(𝐻𝑦)) ↔ (𝑏𝑅𝑎 → (𝐻𝑏)𝑆(𝐻𝑎))))
4035, 39rspc2va 3475 . . . . . . . . . . . 12 (((𝑏𝐴𝑎𝐴) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦))) → (𝑏𝑅𝑎 → (𝐻𝑏)𝑆(𝐻𝑎)))
4140ancoms 450 . . . . . . . . . . 11 ((∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)) ∧ (𝑏𝐴𝑎𝐴)) → (𝑏𝑅𝑎 → (𝐻𝑏)𝑆(𝐻𝑎)))
4241ancom2s 640 . . . . . . . . . 10 ((∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)) ∧ (𝑎𝐴𝑏𝐴)) → (𝑏𝑅𝑎 → (𝐻𝑏)𝑆(𝐻𝑎)))
438, 42sylan 575 . . . . . . . . 9 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → (𝑏𝑅𝑎 → (𝐻𝑏)𝑆(𝐻𝑎)))
44 breq2 4813 . . . . . . . . . . . . 13 ((𝐻𝑎) = (𝐻𝑏) → ((𝐻𝑏)𝑆(𝐻𝑎) ↔ (𝐻𝑏)𝑆(𝐻𝑏)))
4544notbid 309 . . . . . . . . . . . 12 ((𝐻𝑎) = (𝐻𝑏) → (¬ (𝐻𝑏)𝑆(𝐻𝑎) ↔ ¬ (𝐻𝑏)𝑆(𝐻𝑏)))
4625, 45syl5ibrcom 238 . . . . . . . . . . 11 ((𝑆 Po 𝐵 ∧ (𝐻𝑏) ∈ 𝐵) → ((𝐻𝑎) = (𝐻𝑏) → ¬ (𝐻𝑏)𝑆(𝐻𝑎)))
4720, 24, 46syl2anc 579 . . . . . . . . . 10 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → ((𝐻𝑎) = (𝐻𝑏) → ¬ (𝐻𝑏)𝑆(𝐻𝑎)))
4847con2d 131 . . . . . . . . 9 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → ((𝐻𝑏)𝑆(𝐻𝑎) → ¬ (𝐻𝑎) = (𝐻𝑏)))
4943, 48syld 47 . . . . . . . 8 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → (𝑏𝑅𝑎 → ¬ (𝐻𝑎) = (𝐻𝑏)))
5031, 49jaod 885 . . . . . . 7 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → ((𝑎𝑅𝑏𝑏𝑅𝑎) → ¬ (𝐻𝑎) = (𝐻𝑏)))
517, 50sylbird 251 . . . . . 6 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → (¬ 𝑎 = 𝑏 → ¬ (𝐻𝑎) = (𝐻𝑏)))
5251con4d 115 . . . . 5 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → ((𝐻𝑎) = (𝐻𝑏) → 𝑎 = 𝑏))
5352ralrimivva 3118 . . . 4 (((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) → ∀𝑎𝐴𝑏𝐴 ((𝐻𝑎) = (𝐻𝑏) → 𝑎 = 𝑏))
54 dff13 6704 . . . 4 (𝐻:𝐴1-1𝐵 ↔ (𝐻:𝐴𝐵 ∧ ∀𝑎𝐴𝑏𝐴 ((𝐻𝑎) = (𝐻𝑏) → 𝑎 = 𝑏)))
553, 53, 54sylanbrc 578 . . 3 (((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) → 𝐻:𝐴1-1𝐵)
56 df-f1o 6075 . . 3 (𝐻:𝐴1-1-onto𝐵 ↔ (𝐻:𝐴1-1𝐵𝐻:𝐴onto𝐵))
5755, 1, 56sylanbrc 578 . 2 (((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) → 𝐻:𝐴1-1-onto𝐵)
58 sotric 5224 . . . . . . 7 ((𝑅 Or 𝐴 ∧ (𝑎𝐴𝑏𝐴)) → (𝑎𝑅𝑏 ↔ ¬ (𝑎 = 𝑏𝑏𝑅𝑎)))
5958con2bid 345 . . . . . 6 ((𝑅 Or 𝐴 ∧ (𝑎𝐴𝑏𝐴)) → ((𝑎 = 𝑏𝑏𝑅𝑎) ↔ ¬ 𝑎𝑅𝑏))
604, 59sylan 575 . . . . 5 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → ((𝑎 = 𝑏𝑏𝑅𝑎) ↔ ¬ 𝑎𝑅𝑏))
61 fveq2 6375 . . . . . . . . . 10 (𝑎 = 𝑏 → (𝐻𝑎) = (𝐻𝑏))
6261breq1d 4819 . . . . . . . . 9 (𝑎 = 𝑏 → ((𝐻𝑎)𝑆(𝐻𝑏) ↔ (𝐻𝑏)𝑆(𝐻𝑏)))
6362notbid 309 . . . . . . . 8 (𝑎 = 𝑏 → (¬ (𝐻𝑎)𝑆(𝐻𝑏) ↔ ¬ (𝐻𝑏)𝑆(𝐻𝑏)))
6425, 63syl5ibrcom 238 . . . . . . 7 ((𝑆 Po 𝐵 ∧ (𝐻𝑏) ∈ 𝐵) → (𝑎 = 𝑏 → ¬ (𝐻𝑎)𝑆(𝐻𝑏)))
6520, 24, 64syl2anc 579 . . . . . 6 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → (𝑎 = 𝑏 → ¬ (𝐻𝑎)𝑆(𝐻𝑏)))
66 simprl 787 . . . . . . . . 9 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → 𝑎𝐴)
6722, 66ffvelrnd 6550 . . . . . . . 8 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → (𝐻𝑎) ∈ 𝐵)
68 po2nr 5211 . . . . . . . . 9 ((𝑆 Po 𝐵 ∧ ((𝐻𝑏) ∈ 𝐵 ∧ (𝐻𝑎) ∈ 𝐵)) → ¬ ((𝐻𝑏)𝑆(𝐻𝑎) ∧ (𝐻𝑎)𝑆(𝐻𝑏)))
69 imnan 388 . . . . . . . . 9 (((𝐻𝑏)𝑆(𝐻𝑎) → ¬ (𝐻𝑎)𝑆(𝐻𝑏)) ↔ ¬ ((𝐻𝑏)𝑆(𝐻𝑎) ∧ (𝐻𝑎)𝑆(𝐻𝑏)))
7068, 69sylibr 225 . . . . . . . 8 ((𝑆 Po 𝐵 ∧ ((𝐻𝑏) ∈ 𝐵 ∧ (𝐻𝑎) ∈ 𝐵)) → ((𝐻𝑏)𝑆(𝐻𝑎) → ¬ (𝐻𝑎)𝑆(𝐻𝑏)))
7120, 24, 67, 70syl12anc 865 . . . . . . 7 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → ((𝐻𝑏)𝑆(𝐻𝑎) → ¬ (𝐻𝑎)𝑆(𝐻𝑏)))
7243, 71syld 47 . . . . . 6 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → (𝑏𝑅𝑎 → ¬ (𝐻𝑎)𝑆(𝐻𝑏)))
7365, 72jaod 885 . . . . 5 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → ((𝑎 = 𝑏𝑏𝑅𝑎) → ¬ (𝐻𝑎)𝑆(𝐻𝑏)))
7460, 73sylbird 251 . . . 4 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → (¬ 𝑎𝑅𝑏 → ¬ (𝐻𝑎)𝑆(𝐻𝑏)))
7519, 74impcon4bid 218 . . 3 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)))
7675ralrimivva 3118 . 2 (((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) → ∀𝑎𝐴𝑏𝐴 (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)))
77 df-isom 6077 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑎𝐴𝑏𝐴 (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏))))
7857, 76, 77sylanbrc 578 1 (((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 873   = wceq 1652  wcel 2155  wral 3055   class class class wbr 4809   Po wpo 5196   Or wor 5197  wf 6064  1-1wf1 6065  ontowfo 6066  1-1-ontowf1o 6067  cfv 6068   Isom wiso 6069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-id 5185  df-po 5198  df-so 5199  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077
This theorem is referenced by:  ordtypelem8  8637  cantnf  8805  fin23lem27  9403  iccpnfhmeo  23023  xrhmeo  23024  logccv  24700  xrge0iifiso  30428
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