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Theorem wemoiso 7659
Description: Thus, there is at most one isomorphism between any two well-ordered sets. TODO: Shorten finnisoeu 9526. (Contributed by Stefan O'Rear, 12-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
Assertion
Ref Expression
wemoiso (𝑅 We 𝐴 → ∃*𝑓 𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵))
Distinct variable groups:   𝑅,𝑓   𝐴,𝑓   𝑆,𝑓   𝐵,𝑓

Proof of Theorem wemoiso
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 simpl 486 . . . . . 6 ((𝑅 We 𝐴 ∧ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝑅 We 𝐴)
2 vex 3482 . . . . . . . . 9 𝑓 ∈ V
3 isof1o 7060 . . . . . . . . . 10 (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝑓:𝐴1-1-onto𝐵)
4 f1of 6598 . . . . . . . . . 10 (𝑓:𝐴1-1-onto𝐵𝑓:𝐴𝐵)
53, 4syl 17 . . . . . . . . 9 (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝑓:𝐴𝐵)
6 dmfex 7626 . . . . . . . . 9 ((𝑓 ∈ V ∧ 𝑓:𝐴𝐵) → 𝐴 ∈ V)
72, 5, 6sylancr 590 . . . . . . . 8 (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐴 ∈ V)
87ad2antrl 727 . . . . . . 7 ((𝑅 We 𝐴 ∧ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝐴 ∈ V)
9 exse 5502 . . . . . . 7 (𝐴 ∈ V → 𝑅 Se 𝐴)
108, 9syl 17 . . . . . 6 ((𝑅 We 𝐴 ∧ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝑅 Se 𝐴)
111, 10jca 515 . . . . 5 ((𝑅 We 𝐴 ∧ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → (𝑅 We 𝐴𝑅 Se 𝐴))
12 weisoeq 7092 . . . . 5 (((𝑅 We 𝐴𝑅 Se 𝐴) ∧ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝑓 = 𝑔)
1311, 12sylancom 591 . . . 4 ((𝑅 We 𝐴 ∧ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝑓 = 𝑔)
1413ex 416 . . 3 (𝑅 We 𝐴 → ((𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵)) → 𝑓 = 𝑔))
1514alrimivv 1930 . 2 (𝑅 We 𝐴 → ∀𝑓𝑔((𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵)) → 𝑓 = 𝑔))
16 isoeq1 7054 . . 3 (𝑓 = 𝑔 → (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵)))
1716mo4 2651 . 2 (∃*𝑓 𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ ∀𝑓𝑔((𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵)) → 𝑓 = 𝑔))
1815, 17sylibr 237 1 (𝑅 We 𝐴 → ∃*𝑓 𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wal 1536  wcel 2115  ∃*wmo 2622  Vcvv 3479   Se wse 5495   We wwe 5496  wf 6334  1-1-ontowf1o 6337   Isom wiso 6339
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5186  ax-nul 5193  ax-pow 5249  ax-pr 5313  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rmo 3140  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4822  df-br 5050  df-opab 5112  df-mpt 5130  df-id 5443  df-po 5457  df-so 5458  df-fr 5497  df-se 5498  df-we 5499  df-xp 5544  df-rel 5545  df-cnv 5546  df-co 5547  df-dm 5548  df-rn 5549  df-res 5550  df-ima 5551  df-iota 6297  df-fun 6340  df-fn 6341  df-f 6342  df-f1 6343  df-fo 6344  df-f1o 6345  df-fv 6346  df-isom 6347
This theorem is referenced by:  fzisoeu  41789
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