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Theorem wemoiso2 7908
Description: Thus, there is at most one isomorphism between any two well-ordered sets. (Contributed by Stefan O'Rear, 12-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
Assertion
Ref Expression
wemoiso2 (𝑆 We 𝐵 → ∃*𝑓 𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵))
Distinct variable groups:   𝑅,𝑓   𝐴,𝑓   𝑆,𝑓   𝐵,𝑓

Proof of Theorem wemoiso2
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 simpl 484 . . . . . 6 ((𝑆 We 𝐵 ∧ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝑆 We 𝐵)
2 isof1o 7269 . . . . . . . . . 10 (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝑓:𝐴1-1-onto𝐵)
3 f1ofo 6792 . . . . . . . . . 10 (𝑓:𝐴1-1-onto𝐵𝑓:𝐴onto𝐵)
4 forn 6760 . . . . . . . . . 10 (𝑓:𝐴onto𝐵 → ran 𝑓 = 𝐵)
52, 3, 43syl 18 . . . . . . . . 9 (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ran 𝑓 = 𝐵)
6 vex 3448 . . . . . . . . . 10 𝑓 ∈ V
76rnex 7850 . . . . . . . . 9 ran 𝑓 ∈ V
85, 7eqeltrrdi 2843 . . . . . . . 8 (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐵 ∈ V)
98ad2antrl 727 . . . . . . 7 ((𝑆 We 𝐵 ∧ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝐵 ∈ V)
10 exse 5597 . . . . . . 7 (𝐵 ∈ V → 𝑆 Se 𝐵)
119, 10syl 17 . . . . . 6 ((𝑆 We 𝐵 ∧ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝑆 Se 𝐵)
121, 11jca 513 . . . . 5 ((𝑆 We 𝐵 ∧ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → (𝑆 We 𝐵𝑆 Se 𝐵))
13 weisoeq2 7302 . . . . 5 (((𝑆 We 𝐵𝑆 Se 𝐵) ∧ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝑓 = 𝑔)
1412, 13sylancom 589 . . . 4 ((𝑆 We 𝐵 ∧ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝑓 = 𝑔)
1514ex 414 . . 3 (𝑆 We 𝐵 → ((𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵)) → 𝑓 = 𝑔))
1615alrimivv 1932 . 2 (𝑆 We 𝐵 → ∀𝑓𝑔((𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵)) → 𝑓 = 𝑔))
17 isoeq1 7263 . . 3 (𝑓 = 𝑔 → (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵)))
1817mo4 2561 . 2 (∃*𝑓 𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ ∀𝑓𝑔((𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵)) → 𝑓 = 𝑔))
1916, 18sylibr 233 1 (𝑆 We 𝐵 → ∃*𝑓 𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wal 1540   = wceq 1542  wcel 2107  ∃*wmo 2533  Vcvv 3444   Se wse 5587   We wwe 5588  ran crn 5635  ontowfo 6495  1-1-ontowf1o 6496   Isom wiso 6498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506
This theorem is referenced by:  finnisoeu  10054
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