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Theorem weisoeq2 7376
Description: Thus, there is at most one isomorphism between any two set-like well-ordered classes. Class version of wemoiso2 7999. (Contributed by Mario Carneiro, 25-Jun-2015.)
Assertion
Ref Expression
weisoeq2 (((𝑆 We 𝐵𝑆 Se 𝐵) ∧ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝐹 = 𝐺)

Proof of Theorem weisoeq2
StepHypRef Expression
1 isocnv 7350 . . . 4 (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐹 Isom 𝑆, 𝑅 (𝐵, 𝐴))
2 isocnv 7350 . . . 4 (𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐺 Isom 𝑆, 𝑅 (𝐵, 𝐴))
31, 2anim12i 613 . . 3 ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵)) → (𝐹 Isom 𝑆, 𝑅 (𝐵, 𝐴) ∧ 𝐺 Isom 𝑆, 𝑅 (𝐵, 𝐴)))
4 weisoeq 7375 . . 3 (((𝑆 We 𝐵𝑆 Se 𝐵) ∧ (𝐹 Isom 𝑆, 𝑅 (𝐵, 𝐴) ∧ 𝐺 Isom 𝑆, 𝑅 (𝐵, 𝐴))) → 𝐹 = 𝐺)
53, 4sylan2 593 . 2 (((𝑆 We 𝐵𝑆 Se 𝐵) ∧ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝐹 = 𝐺)
6 simprl 771 . . . 4 (((𝑆 We 𝐵𝑆 Se 𝐵) ∧ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))
7 isof1o 7343 . . . 4 (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐹:𝐴1-1-onto𝐵)
8 f1orel 6851 . . . 4 (𝐹:𝐴1-1-onto𝐵 → Rel 𝐹)
96, 7, 83syl 18 . . 3 (((𝑆 We 𝐵𝑆 Se 𝐵) ∧ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → Rel 𝐹)
10 simprr 773 . . . 4 (((𝑆 We 𝐵𝑆 Se 𝐵) ∧ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))
11 isof1o 7343 . . . 4 (𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐺:𝐴1-1-onto𝐵)
12 f1orel 6851 . . . 4 (𝐺:𝐴1-1-onto𝐵 → Rel 𝐺)
1310, 11, 123syl 18 . . 3 (((𝑆 We 𝐵𝑆 Se 𝐵) ∧ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → Rel 𝐺)
14 cnveqb 6216 . . 3 ((Rel 𝐹 ∧ Rel 𝐺) → (𝐹 = 𝐺𝐹 = 𝐺))
159, 13, 14syl2anc 584 . 2 (((𝑆 We 𝐵𝑆 Se 𝐵) ∧ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → (𝐹 = 𝐺𝐹 = 𝐺))
165, 15mpbird 257 1 (((𝑆 We 𝐵𝑆 Se 𝐵) ∧ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝐹 = 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540   Se wse 5635   We wwe 5636  ccnv 5684  Rel wrel 5690  1-1-ontowf1o 6560   Isom wiso 6562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570
This theorem is referenced by:  wemoiso2  7999
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