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Theorem isopolem 6495
Description: Lemma for isopo 6496. (Contributed by Stefan O'Rear, 16-Nov-2014.)
Assertion
Ref Expression
isopolem (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Po 𝐵𝑅 Po 𝐴))

Proof of Theorem isopolem
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isof1o 6473 . . . . . . . . . . 11 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1-onto𝐵)
2 f1of 6050 . . . . . . . . . . 11 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴𝐵)
3 ffvelrn 6265 . . . . . . . . . . . . 13 ((𝐻:𝐴𝐵𝑑𝐴) → (𝐻𝑑) ∈ 𝐵)
43ex 449 . . . . . . . . . . . 12 (𝐻:𝐴𝐵 → (𝑑𝐴 → (𝐻𝑑) ∈ 𝐵))
5 ffvelrn 6265 . . . . . . . . . . . . 13 ((𝐻:𝐴𝐵𝑒𝐴) → (𝐻𝑒) ∈ 𝐵)
65ex 449 . . . . . . . . . . . 12 (𝐻:𝐴𝐵 → (𝑒𝐴 → (𝐻𝑒) ∈ 𝐵))
7 ffvelrn 6265 . . . . . . . . . . . . 13 ((𝐻:𝐴𝐵𝑓𝐴) → (𝐻𝑓) ∈ 𝐵)
87ex 449 . . . . . . . . . . . 12 (𝐻:𝐴𝐵 → (𝑓𝐴 → (𝐻𝑓) ∈ 𝐵))
94, 6, 83anim123d 1398 . . . . . . . . . . 11 (𝐻:𝐴𝐵 → ((𝑑𝐴𝑒𝐴𝑓𝐴) → ((𝐻𝑑) ∈ 𝐵 ∧ (𝐻𝑒) ∈ 𝐵 ∧ (𝐻𝑓) ∈ 𝐵)))
101, 2, 93syl 18 . . . . . . . . . 10 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ((𝑑𝐴𝑒𝐴𝑓𝐴) → ((𝐻𝑑) ∈ 𝐵 ∧ (𝐻𝑒) ∈ 𝐵 ∧ (𝐻𝑓) ∈ 𝐵)))
1110imp 444 . . . . . . . . 9 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑒𝐴𝑓𝐴)) → ((𝐻𝑑) ∈ 𝐵 ∧ (𝐻𝑒) ∈ 𝐵 ∧ (𝐻𝑓) ∈ 𝐵))
12 breq12 4588 . . . . . . . . . . . . 13 ((𝑎 = (𝐻𝑑) ∧ 𝑎 = (𝐻𝑑)) → (𝑎𝑆𝑎 ↔ (𝐻𝑑)𝑆(𝐻𝑑)))
1312anidms 675 . . . . . . . . . . . 12 (𝑎 = (𝐻𝑑) → (𝑎𝑆𝑎 ↔ (𝐻𝑑)𝑆(𝐻𝑑)))
1413notbid 307 . . . . . . . . . . 11 (𝑎 = (𝐻𝑑) → (¬ 𝑎𝑆𝑎 ↔ ¬ (𝐻𝑑)𝑆(𝐻𝑑)))
15 breq1 4586 . . . . . . . . . . . . 13 (𝑎 = (𝐻𝑑) → (𝑎𝑆𝑏 ↔ (𝐻𝑑)𝑆𝑏))
1615anbi1d 737 . . . . . . . . . . . 12 (𝑎 = (𝐻𝑑) → ((𝑎𝑆𝑏𝑏𝑆𝑐) ↔ ((𝐻𝑑)𝑆𝑏𝑏𝑆𝑐)))
17 breq1 4586 . . . . . . . . . . . 12 (𝑎 = (𝐻𝑑) → (𝑎𝑆𝑐 ↔ (𝐻𝑑)𝑆𝑐))
1816, 17imbi12d 333 . . . . . . . . . . 11 (𝑎 = (𝐻𝑑) → (((𝑎𝑆𝑏𝑏𝑆𝑐) → 𝑎𝑆𝑐) ↔ (((𝐻𝑑)𝑆𝑏𝑏𝑆𝑐) → (𝐻𝑑)𝑆𝑐)))
1914, 18anbi12d 743 . . . . . . . . . 10 (𝑎 = (𝐻𝑑) → ((¬ 𝑎𝑆𝑎 ∧ ((𝑎𝑆𝑏𝑏𝑆𝑐) → 𝑎𝑆𝑐)) ↔ (¬ (𝐻𝑑)𝑆(𝐻𝑑) ∧ (((𝐻𝑑)𝑆𝑏𝑏𝑆𝑐) → (𝐻𝑑)𝑆𝑐))))
20 breq2 4587 . . . . . . . . . . . . 13 (𝑏 = (𝐻𝑒) → ((𝐻𝑑)𝑆𝑏 ↔ (𝐻𝑑)𝑆(𝐻𝑒)))
21 breq1 4586 . . . . . . . . . . . . 13 (𝑏 = (𝐻𝑒) → (𝑏𝑆𝑐 ↔ (𝐻𝑒)𝑆𝑐))
2220, 21anbi12d 743 . . . . . . . . . . . 12 (𝑏 = (𝐻𝑒) → (((𝐻𝑑)𝑆𝑏𝑏𝑆𝑐) ↔ ((𝐻𝑑)𝑆(𝐻𝑒) ∧ (𝐻𝑒)𝑆𝑐)))
2322imbi1d 330 . . . . . . . . . . 11 (𝑏 = (𝐻𝑒) → ((((𝐻𝑑)𝑆𝑏𝑏𝑆𝑐) → (𝐻𝑑)𝑆𝑐) ↔ (((𝐻𝑑)𝑆(𝐻𝑒) ∧ (𝐻𝑒)𝑆𝑐) → (𝐻𝑑)𝑆𝑐)))
2423anbi2d 736 . . . . . . . . . 10 (𝑏 = (𝐻𝑒) → ((¬ (𝐻𝑑)𝑆(𝐻𝑑) ∧ (((𝐻𝑑)𝑆𝑏𝑏𝑆𝑐) → (𝐻𝑑)𝑆𝑐)) ↔ (¬ (𝐻𝑑)𝑆(𝐻𝑑) ∧ (((𝐻𝑑)𝑆(𝐻𝑒) ∧ (𝐻𝑒)𝑆𝑐) → (𝐻𝑑)𝑆𝑐))))
25 breq2 4587 . . . . . . . . . . . . 13 (𝑐 = (𝐻𝑓) → ((𝐻𝑒)𝑆𝑐 ↔ (𝐻𝑒)𝑆(𝐻𝑓)))
2625anbi2d 736 . . . . . . . . . . . 12 (𝑐 = (𝐻𝑓) → (((𝐻𝑑)𝑆(𝐻𝑒) ∧ (𝐻𝑒)𝑆𝑐) ↔ ((𝐻𝑑)𝑆(𝐻𝑒) ∧ (𝐻𝑒)𝑆(𝐻𝑓))))
27 breq2 4587 . . . . . . . . . . . 12 (𝑐 = (𝐻𝑓) → ((𝐻𝑑)𝑆𝑐 ↔ (𝐻𝑑)𝑆(𝐻𝑓)))
2826, 27imbi12d 333 . . . . . . . . . . 11 (𝑐 = (𝐻𝑓) → ((((𝐻𝑑)𝑆(𝐻𝑒) ∧ (𝐻𝑒)𝑆𝑐) → (𝐻𝑑)𝑆𝑐) ↔ (((𝐻𝑑)𝑆(𝐻𝑒) ∧ (𝐻𝑒)𝑆(𝐻𝑓)) → (𝐻𝑑)𝑆(𝐻𝑓))))
2928anbi2d 736 . . . . . . . . . 10 (𝑐 = (𝐻𝑓) → ((¬ (𝐻𝑑)𝑆(𝐻𝑑) ∧ (((𝐻𝑑)𝑆(𝐻𝑒) ∧ (𝐻𝑒)𝑆𝑐) → (𝐻𝑑)𝑆𝑐)) ↔ (¬ (𝐻𝑑)𝑆(𝐻𝑑) ∧ (((𝐻𝑑)𝑆(𝐻𝑒) ∧ (𝐻𝑒)𝑆(𝐻𝑓)) → (𝐻𝑑)𝑆(𝐻𝑓)))))
3019, 24, 29rspc3v 3296 . . . . . . . . 9 (((𝐻𝑑) ∈ 𝐵 ∧ (𝐻𝑒) ∈ 𝐵 ∧ (𝐻𝑓) ∈ 𝐵) → (∀𝑎𝐵𝑏𝐵𝑐𝐵𝑎𝑆𝑎 ∧ ((𝑎𝑆𝑏𝑏𝑆𝑐) → 𝑎𝑆𝑐)) → (¬ (𝐻𝑑)𝑆(𝐻𝑑) ∧ (((𝐻𝑑)𝑆(𝐻𝑒) ∧ (𝐻𝑒)𝑆(𝐻𝑓)) → (𝐻𝑑)𝑆(𝐻𝑓)))))
3111, 30syl 17 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑒𝐴𝑓𝐴)) → (∀𝑎𝐵𝑏𝐵𝑐𝐵𝑎𝑆𝑎 ∧ ((𝑎𝑆𝑏𝑏𝑆𝑐) → 𝑎𝑆𝑐)) → (¬ (𝐻𝑑)𝑆(𝐻𝑑) ∧ (((𝐻𝑑)𝑆(𝐻𝑒) ∧ (𝐻𝑒)𝑆(𝐻𝑓)) → (𝐻𝑑)𝑆(𝐻𝑓)))))
32 simpl 472 . . . . . . . . . . 11 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑒𝐴𝑓𝐴)) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
33 simpr1 1060 . . . . . . . . . . 11 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑒𝐴𝑓𝐴)) → 𝑑𝐴)
34 isorel 6476 . . . . . . . . . . 11 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑑𝐴)) → (𝑑𝑅𝑑 ↔ (𝐻𝑑)𝑆(𝐻𝑑)))
3532, 33, 33, 34syl12anc 1316 . . . . . . . . . 10 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑒𝐴𝑓𝐴)) → (𝑑𝑅𝑑 ↔ (𝐻𝑑)𝑆(𝐻𝑑)))
3635notbid 307 . . . . . . . . 9 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑒𝐴𝑓𝐴)) → (¬ 𝑑𝑅𝑑 ↔ ¬ (𝐻𝑑)𝑆(𝐻𝑑)))
37 simpr2 1061 . . . . . . . . . . . 12 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑒𝐴𝑓𝐴)) → 𝑒𝐴)
38 isorel 6476 . . . . . . . . . . . 12 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑒𝐴)) → (𝑑𝑅𝑒 ↔ (𝐻𝑑)𝑆(𝐻𝑒)))
3932, 33, 37, 38syl12anc 1316 . . . . . . . . . . 11 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑒𝐴𝑓𝐴)) → (𝑑𝑅𝑒 ↔ (𝐻𝑑)𝑆(𝐻𝑒)))
40 simpr3 1062 . . . . . . . . . . . 12 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑒𝐴𝑓𝐴)) → 𝑓𝐴)
41 isorel 6476 . . . . . . . . . . . 12 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑒𝐴𝑓𝐴)) → (𝑒𝑅𝑓 ↔ (𝐻𝑒)𝑆(𝐻𝑓)))
4232, 37, 40, 41syl12anc 1316 . . . . . . . . . . 11 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑒𝐴𝑓𝐴)) → (𝑒𝑅𝑓 ↔ (𝐻𝑒)𝑆(𝐻𝑓)))
4339, 42anbi12d 743 . . . . . . . . . 10 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑒𝐴𝑓𝐴)) → ((𝑑𝑅𝑒𝑒𝑅𝑓) ↔ ((𝐻𝑑)𝑆(𝐻𝑒) ∧ (𝐻𝑒)𝑆(𝐻𝑓))))
44 isorel 6476 . . . . . . . . . . 11 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑓𝐴)) → (𝑑𝑅𝑓 ↔ (𝐻𝑑)𝑆(𝐻𝑓)))
4532, 33, 40, 44syl12anc 1316 . . . . . . . . . 10 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑒𝐴𝑓𝐴)) → (𝑑𝑅𝑓 ↔ (𝐻𝑑)𝑆(𝐻𝑓)))
4643, 45imbi12d 333 . . . . . . . . 9 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑒𝐴𝑓𝐴)) → (((𝑑𝑅𝑒𝑒𝑅𝑓) → 𝑑𝑅𝑓) ↔ (((𝐻𝑑)𝑆(𝐻𝑒) ∧ (𝐻𝑒)𝑆(𝐻𝑓)) → (𝐻𝑑)𝑆(𝐻𝑓))))
4736, 46anbi12d 743 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑒𝐴𝑓𝐴)) → ((¬ 𝑑𝑅𝑑 ∧ ((𝑑𝑅𝑒𝑒𝑅𝑓) → 𝑑𝑅𝑓)) ↔ (¬ (𝐻𝑑)𝑆(𝐻𝑑) ∧ (((𝐻𝑑)𝑆(𝐻𝑒) ∧ (𝐻𝑒)𝑆(𝐻𝑓)) → (𝐻𝑑)𝑆(𝐻𝑓)))))
4831, 47sylibrd 248 . . . . . . 7 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑𝐴𝑒𝐴𝑓𝐴)) → (∀𝑎𝐵𝑏𝐵𝑐𝐵𝑎𝑆𝑎 ∧ ((𝑎𝑆𝑏𝑏𝑆𝑐) → 𝑎𝑆𝑐)) → (¬ 𝑑𝑅𝑑 ∧ ((𝑑𝑅𝑒𝑒𝑅𝑓) → 𝑑𝑅𝑓))))
4948ex 449 . . . . . 6 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ((𝑑𝐴𝑒𝐴𝑓𝐴) → (∀𝑎𝐵𝑏𝐵𝑐𝐵𝑎𝑆𝑎 ∧ ((𝑎𝑆𝑏𝑏𝑆𝑐) → 𝑎𝑆𝑐)) → (¬ 𝑑𝑅𝑑 ∧ ((𝑑𝑅𝑒𝑒𝑅𝑓) → 𝑑𝑅𝑓)))))
5049com23 84 . . . . 5 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (∀𝑎𝐵𝑏𝐵𝑐𝐵𝑎𝑆𝑎 ∧ ((𝑎𝑆𝑏𝑏𝑆𝑐) → 𝑎𝑆𝑐)) → ((𝑑𝐴𝑒𝐴𝑓𝐴) → (¬ 𝑑𝑅𝑑 ∧ ((𝑑𝑅𝑒𝑒𝑅𝑓) → 𝑑𝑅𝑓)))))
5150imp31 447 . . . 4 (((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ∀𝑎𝐵𝑏𝐵𝑐𝐵𝑎𝑆𝑎 ∧ ((𝑎𝑆𝑏𝑏𝑆𝑐) → 𝑎𝑆𝑐))) ∧ (𝑑𝐴𝑒𝐴𝑓𝐴)) → (¬ 𝑑𝑅𝑑 ∧ ((𝑑𝑅𝑒𝑒𝑅𝑓) → 𝑑𝑅𝑓)))
5251ralrimivvva 2955 . . 3 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ∀𝑎𝐵𝑏𝐵𝑐𝐵𝑎𝑆𝑎 ∧ ((𝑎𝑆𝑏𝑏𝑆𝑐) → 𝑎𝑆𝑐))) → ∀𝑑𝐴𝑒𝐴𝑓𝐴𝑑𝑅𝑑 ∧ ((𝑑𝑅𝑒𝑒𝑅𝑓) → 𝑑𝑅𝑓)))
5352ex 449 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (∀𝑎𝐵𝑏𝐵𝑐𝐵𝑎𝑆𝑎 ∧ ((𝑎𝑆𝑏𝑏𝑆𝑐) → 𝑎𝑆𝑐)) → ∀𝑑𝐴𝑒𝐴𝑓𝐴𝑑𝑅𝑑 ∧ ((𝑑𝑅𝑒𝑒𝑅𝑓) → 𝑑𝑅𝑓))))
54 df-po 4959 . 2 (𝑆 Po 𝐵 ↔ ∀𝑎𝐵𝑏𝐵𝑐𝐵𝑎𝑆𝑎 ∧ ((𝑎𝑆𝑏𝑏𝑆𝑐) → 𝑎𝑆𝑐)))
55 df-po 4959 . 2 (𝑅 Po 𝐴 ↔ ∀𝑑𝐴𝑒𝐴𝑓𝐴𝑑𝑅𝑑 ∧ ((𝑑𝑅𝑒𝑒𝑅𝑓) → 𝑑𝑅𝑓)))
5653, 54, 553imtr4g 284 1 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Po 𝐵𝑅 Po 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wral 2896   class class class wbr 4583   Po wpo 4957  wf 5800  1-1-ontowf1o 5803  cfv 5804   Isom wiso 5805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-po 4959  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-f1o 5811  df-fv 5812  df-isom 5813
This theorem is referenced by:  isopo  6496  isosolem  6497
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