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Theorem weisoeq2 6563
Description: Thus, there is at most one isomorphism between any two set-like well-ordered classes. Class version of wemoiso2 7102. (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 6537 . . . 4 (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐹 Isom 𝑆, 𝑅 (𝐵, 𝐴))
2 isocnv 6537 . . . 4 (𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐺 Isom 𝑆, 𝑅 (𝐵, 𝐴))
31, 2anim12i 589 . . 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 6562 . . 3 (((𝑆 We 𝐵𝑆 Se 𝐵) ∧ (𝐹 Isom 𝑆, 𝑅 (𝐵, 𝐴) ∧ 𝐺 Isom 𝑆, 𝑅 (𝐵, 𝐴))) → 𝐹 = 𝐺)
53, 4sylan2 491 . 2 (((𝑆 We 𝐵𝑆 Se 𝐵) ∧ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝐹 = 𝐺)
6 simprl 793 . . . 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 6530 . . . 4 (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐹:𝐴1-1-onto𝐵)
8 f1orel 6099 . . . 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 795 . . . 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 6530 . . . 4 (𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐺:𝐴1-1-onto𝐵)
12 f1orel 6099 . . . 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 5551 . . 3 ((Rel 𝐹 ∧ Rel 𝐺) → (𝐹 = 𝐺𝐹 = 𝐺))
159, 13, 14syl2anc 692 . 2 (((𝑆 We 𝐵𝑆 Se 𝐵) ∧ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → (𝐹 = 𝐺𝐹 = 𝐺))
165, 15mpbird 247 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 196  wa 384   = wceq 1480   Se wse 5033   We wwe 5034  ccnv 5075  Rel wrel 5081  1-1-ontowf1o 5848   Isom wiso 5850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-se 5036  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-isom 5858
This theorem is referenced by:  wemoiso2  7102
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