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Theorem soisoi 7273
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 769 . . . . 5 (((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) → 𝐻:𝐴onto𝐵)
2 fof 6756 . . . . 5 (𝐻:𝐴onto𝐵𝐻:𝐴𝐵)
31, 2syl 17 . . . 4 (((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) → 𝐻:𝐴𝐵)
4 sotrieq 5574 . . . . . . . . 9 ((𝑅 Or 𝐴 ∧ (𝑎𝐴𝑏𝐴)) → (𝑎 = 𝑏 ↔ ¬ (𝑎𝑅𝑏𝑏𝑅𝑎)))
54con2bid 354 . . . . . . . 8 ((𝑅 Or 𝐴 ∧ (𝑎𝐴𝑏𝐴)) → ((𝑎𝑅𝑏𝑏𝑅𝑎) ↔ ¬ 𝑎 = 𝑏))
65ad4ant14 750 . . . . . . 7 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → ((𝑎𝑅𝑏𝑏𝑅𝑎) ↔ ¬ 𝑎 = 𝑏))
7 simprr 771 . . . . . . . . . 10 (((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) → ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))
8 breq1 5108 . . . . . . . . . . . . 13 (𝑥 = 𝑎 → (𝑥𝑅𝑦𝑎𝑅𝑦))
9 fveq2 6842 . . . . . . . . . . . . . 14 (𝑥 = 𝑎 → (𝐻𝑥) = (𝐻𝑎))
109breq1d 5115 . . . . . . . . . . . . 13 (𝑥 = 𝑎 → ((𝐻𝑥)𝑆(𝐻𝑦) ↔ (𝐻𝑎)𝑆(𝐻𝑦)))
118, 10imbi12d 344 . . . . . . . . . . . 12 (𝑥 = 𝑎 → ((𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)) ↔ (𝑎𝑅𝑦 → (𝐻𝑎)𝑆(𝐻𝑦))))
12 breq2 5109 . . . . . . . . . . . . 13 (𝑦 = 𝑏 → (𝑎𝑅𝑦𝑎𝑅𝑏))
13 fveq2 6842 . . . . . . . . . . . . . 14 (𝑦 = 𝑏 → (𝐻𝑦) = (𝐻𝑏))
1413breq2d 5117 . . . . . . . . . . . . 13 (𝑦 = 𝑏 → ((𝐻𝑎)𝑆(𝐻𝑦) ↔ (𝐻𝑎)𝑆(𝐻𝑏)))
1512, 14imbi12d 344 . . . . . . . . . . . 12 (𝑦 = 𝑏 → ((𝑎𝑅𝑦 → (𝐻𝑎)𝑆(𝐻𝑦)) ↔ (𝑎𝑅𝑏 → (𝐻𝑎)𝑆(𝐻𝑏))))
1611, 15rspc2va 3591 . . . . . . . . . . 11 (((𝑎𝐴𝑏𝐴) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦))) → (𝑎𝑅𝑏 → (𝐻𝑎)𝑆(𝐻𝑏)))
1716ancoms 459 . . . . . . . . . 10 ((∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)) ∧ (𝑎𝐴𝑏𝐴)) → (𝑎𝑅𝑏 → (𝐻𝑎)𝑆(𝐻𝑏)))
187, 17sylan 580 . . . . . . . . 9 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → (𝑎𝑅𝑏 → (𝐻𝑎)𝑆(𝐻𝑏)))
19 simpllr 774 . . . . . . . . . . 11 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → 𝑆 Po 𝐵)
20 simplrl 775 . . . . . . . . . . . . 13 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → 𝐻:𝐴onto𝐵)
2120, 2syl 17 . . . . . . . . . . . 12 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → 𝐻:𝐴𝐵)
22 simprr 771 . . . . . . . . . . . 12 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → 𝑏𝐴)
2321, 22ffvelcdmd 7036 . . . . . . . . . . 11 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → (𝐻𝑏) ∈ 𝐵)
24 poirr 5557 . . . . . . . . . . . 12 ((𝑆 Po 𝐵 ∧ (𝐻𝑏) ∈ 𝐵) → ¬ (𝐻𝑏)𝑆(𝐻𝑏))
25 breq1 5108 . . . . . . . . . . . . 13 ((𝐻𝑎) = (𝐻𝑏) → ((𝐻𝑎)𝑆(𝐻𝑏) ↔ (𝐻𝑏)𝑆(𝐻𝑏)))
2625notbid 317 . . . . . . . . . . . 12 ((𝐻𝑎) = (𝐻𝑏) → (¬ (𝐻𝑎)𝑆(𝐻𝑏) ↔ ¬ (𝐻𝑏)𝑆(𝐻𝑏)))
2724, 26syl5ibrcom 246 . . . . . . . . . . 11 ((𝑆 Po 𝐵 ∧ (𝐻𝑏) ∈ 𝐵) → ((𝐻𝑎) = (𝐻𝑏) → ¬ (𝐻𝑎)𝑆(𝐻𝑏)))
2819, 23, 27syl2anc 584 . . . . . . . . . 10 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → ((𝐻𝑎) = (𝐻𝑏) → ¬ (𝐻𝑎)𝑆(𝐻𝑏)))
2928con2d 134 . . . . . . . . 9 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → ((𝐻𝑎)𝑆(𝐻𝑏) → ¬ (𝐻𝑎) = (𝐻𝑏)))
3018, 29syld 47 . . . . . . . 8 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → (𝑎𝑅𝑏 → ¬ (𝐻𝑎) = (𝐻𝑏)))
31 breq1 5108 . . . . . . . . . . . . . 14 (𝑥 = 𝑏 → (𝑥𝑅𝑦𝑏𝑅𝑦))
32 fveq2 6842 . . . . . . . . . . . . . . 15 (𝑥 = 𝑏 → (𝐻𝑥) = (𝐻𝑏))
3332breq1d 5115 . . . . . . . . . . . . . 14 (𝑥 = 𝑏 → ((𝐻𝑥)𝑆(𝐻𝑦) ↔ (𝐻𝑏)𝑆(𝐻𝑦)))
3431, 33imbi12d 344 . . . . . . . . . . . . 13 (𝑥 = 𝑏 → ((𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)) ↔ (𝑏𝑅𝑦 → (𝐻𝑏)𝑆(𝐻𝑦))))
35 breq2 5109 . . . . . . . . . . . . . 14 (𝑦 = 𝑎 → (𝑏𝑅𝑦𝑏𝑅𝑎))
36 fveq2 6842 . . . . . . . . . . . . . . 15 (𝑦 = 𝑎 → (𝐻𝑦) = (𝐻𝑎))
3736breq2d 5117 . . . . . . . . . . . . . 14 (𝑦 = 𝑎 → ((𝐻𝑏)𝑆(𝐻𝑦) ↔ (𝐻𝑏)𝑆(𝐻𝑎)))
3835, 37imbi12d 344 . . . . . . . . . . . . 13 (𝑦 = 𝑎 → ((𝑏𝑅𝑦 → (𝐻𝑏)𝑆(𝐻𝑦)) ↔ (𝑏𝑅𝑎 → (𝐻𝑏)𝑆(𝐻𝑎))))
3934, 38rspc2va 3591 . . . . . . . . . . . 12 (((𝑏𝐴𝑎𝐴) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦))) → (𝑏𝑅𝑎 → (𝐻𝑏)𝑆(𝐻𝑎)))
4039ancoms 459 . . . . . . . . . . 11 ((∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)) ∧ (𝑏𝐴𝑎𝐴)) → (𝑏𝑅𝑎 → (𝐻𝑏)𝑆(𝐻𝑎)))
4140ancom2s 648 . . . . . . . . . 10 ((∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)) ∧ (𝑎𝐴𝑏𝐴)) → (𝑏𝑅𝑎 → (𝐻𝑏)𝑆(𝐻𝑎)))
427, 41sylan 580 . . . . . . . . 9 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → (𝑏𝑅𝑎 → (𝐻𝑏)𝑆(𝐻𝑎)))
43 breq2 5109 . . . . . . . . . . . . 13 ((𝐻𝑎) = (𝐻𝑏) → ((𝐻𝑏)𝑆(𝐻𝑎) ↔ (𝐻𝑏)𝑆(𝐻𝑏)))
4443notbid 317 . . . . . . . . . . . 12 ((𝐻𝑎) = (𝐻𝑏) → (¬ (𝐻𝑏)𝑆(𝐻𝑎) ↔ ¬ (𝐻𝑏)𝑆(𝐻𝑏)))
4524, 44syl5ibrcom 246 . . . . . . . . . . 11 ((𝑆 Po 𝐵 ∧ (𝐻𝑏) ∈ 𝐵) → ((𝐻𝑎) = (𝐻𝑏) → ¬ (𝐻𝑏)𝑆(𝐻𝑎)))
4619, 23, 45syl2anc 584 . . . . . . . . . 10 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → ((𝐻𝑎) = (𝐻𝑏) → ¬ (𝐻𝑏)𝑆(𝐻𝑎)))
4746con2d 134 . . . . . . . . 9 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → ((𝐻𝑏)𝑆(𝐻𝑎) → ¬ (𝐻𝑎) = (𝐻𝑏)))
4842, 47syld 47 . . . . . . . 8 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → (𝑏𝑅𝑎 → ¬ (𝐻𝑎) = (𝐻𝑏)))
4930, 48jaod 857 . . . . . . 7 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → ((𝑎𝑅𝑏𝑏𝑅𝑎) → ¬ (𝐻𝑎) = (𝐻𝑏)))
506, 49sylbird 259 . . . . . 6 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → (¬ 𝑎 = 𝑏 → ¬ (𝐻𝑎) = (𝐻𝑏)))
5150con4d 115 . . . . 5 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → ((𝐻𝑎) = (𝐻𝑏) → 𝑎 = 𝑏))
5251ralrimivva 3197 . . . 4 (((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) → ∀𝑎𝐴𝑏𝐴 ((𝐻𝑎) = (𝐻𝑏) → 𝑎 = 𝑏))
53 dff13 7202 . . . 4 (𝐻:𝐴1-1𝐵 ↔ (𝐻:𝐴𝐵 ∧ ∀𝑎𝐴𝑏𝐴 ((𝐻𝑎) = (𝐻𝑏) → 𝑎 = 𝑏)))
543, 52, 53sylanbrc 583 . . 3 (((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) → 𝐻:𝐴1-1𝐵)
55 df-f1o 6503 . . 3 (𝐻:𝐴1-1-onto𝐵 ↔ (𝐻:𝐴1-1𝐵𝐻:𝐴onto𝐵))
5654, 1, 55sylanbrc 583 . 2 (((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) → 𝐻:𝐴1-1-onto𝐵)
57 sotric 5573 . . . . . . 7 ((𝑅 Or 𝐴 ∧ (𝑎𝐴𝑏𝐴)) → (𝑎𝑅𝑏 ↔ ¬ (𝑎 = 𝑏𝑏𝑅𝑎)))
5857con2bid 354 . . . . . 6 ((𝑅 Or 𝐴 ∧ (𝑎𝐴𝑏𝐴)) → ((𝑎 = 𝑏𝑏𝑅𝑎) ↔ ¬ 𝑎𝑅𝑏))
5958ad4ant14 750 . . . . 5 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → ((𝑎 = 𝑏𝑏𝑅𝑎) ↔ ¬ 𝑎𝑅𝑏))
60 fveq2 6842 . . . . . . . . . 10 (𝑎 = 𝑏 → (𝐻𝑎) = (𝐻𝑏))
6160breq1d 5115 . . . . . . . . 9 (𝑎 = 𝑏 → ((𝐻𝑎)𝑆(𝐻𝑏) ↔ (𝐻𝑏)𝑆(𝐻𝑏)))
6261notbid 317 . . . . . . . 8 (𝑎 = 𝑏 → (¬ (𝐻𝑎)𝑆(𝐻𝑏) ↔ ¬ (𝐻𝑏)𝑆(𝐻𝑏)))
6324, 62syl5ibrcom 246 . . . . . . 7 ((𝑆 Po 𝐵 ∧ (𝐻𝑏) ∈ 𝐵) → (𝑎 = 𝑏 → ¬ (𝐻𝑎)𝑆(𝐻𝑏)))
6419, 23, 63syl2anc 584 . . . . . 6 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → (𝑎 = 𝑏 → ¬ (𝐻𝑎)𝑆(𝐻𝑏)))
65 simprl 769 . . . . . . . . 9 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → 𝑎𝐴)
6621, 65ffvelcdmd 7036 . . . . . . . 8 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → (𝐻𝑎) ∈ 𝐵)
67 po2nr 5559 . . . . . . . . 9 ((𝑆 Po 𝐵 ∧ ((𝐻𝑏) ∈ 𝐵 ∧ (𝐻𝑎) ∈ 𝐵)) → ¬ ((𝐻𝑏)𝑆(𝐻𝑎) ∧ (𝐻𝑎)𝑆(𝐻𝑏)))
68 imnan 400 . . . . . . . . 9 (((𝐻𝑏)𝑆(𝐻𝑎) → ¬ (𝐻𝑎)𝑆(𝐻𝑏)) ↔ ¬ ((𝐻𝑏)𝑆(𝐻𝑎) ∧ (𝐻𝑎)𝑆(𝐻𝑏)))
6967, 68sylibr 233 . . . . . . . 8 ((𝑆 Po 𝐵 ∧ ((𝐻𝑏) ∈ 𝐵 ∧ (𝐻𝑎) ∈ 𝐵)) → ((𝐻𝑏)𝑆(𝐻𝑎) → ¬ (𝐻𝑎)𝑆(𝐻𝑏)))
7019, 23, 66, 69syl12anc 835 . . . . . . 7 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → ((𝐻𝑏)𝑆(𝐻𝑎) → ¬ (𝐻𝑎)𝑆(𝐻𝑏)))
7142, 70syld 47 . . . . . 6 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → (𝑏𝑅𝑎 → ¬ (𝐻𝑎)𝑆(𝐻𝑏)))
7264, 71jaod 857 . . . . 5 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → ((𝑎 = 𝑏𝑏𝑅𝑎) → ¬ (𝐻𝑎)𝑆(𝐻𝑏)))
7359, 72sylbird 259 . . . 4 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → (¬ 𝑎𝑅𝑏 → ¬ (𝐻𝑎)𝑆(𝐻𝑏)))
7418, 73impcon4bid 226 . . 3 ((((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) ∧ (𝑎𝐴𝑏𝐴)) → (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)))
7574ralrimivva 3197 . 2 (((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) → ∀𝑎𝐴𝑏𝐴 (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)))
76 df-isom 6505 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑎𝐴𝑏𝐴 (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏))))
7756, 75, 76sylanbrc 583 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 205  wa 396  wo 845   = wceq 1541  wcel 2106  wral 3064   class class class wbr 5105   Po wpo 5543   Or wor 5544  wf 6492  1-1wf1 6493  ontowfo 6494  1-1-ontowf1o 6495  cfv 6496   Isom wiso 6497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-id 5531  df-po 5545  df-so 5546  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505
This theorem is referenced by:  ordtypelem8  9461  cantnf  9629  fin23lem27  10264  iccpnfhmeo  24308  xrhmeo  24309  logccv  26018  xrge0iifiso  32516
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