MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wemoiso Structured version   Visualization version   GIF version

Theorem wemoiso 7414
Description: Thus, there is at most one isomorphism between any two well-ordered sets. TODO: Shorten finnisoeu 9250. (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 476 . . . . . 6 ((𝑅 We 𝐴 ∧ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝑅 We 𝐴)
2 vex 3418 . . . . . . . . 9 𝑓 ∈ V
3 isof1o 6829 . . . . . . . . . 10 (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝑓:𝐴1-1-onto𝐵)
4 f1of 6379 . . . . . . . . . 10 (𝑓:𝐴1-1-onto𝐵𝑓:𝐴𝐵)
53, 4syl 17 . . . . . . . . 9 (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝑓:𝐴𝐵)
6 dmfex 7387 . . . . . . . . 9 ((𝑓 ∈ V ∧ 𝑓:𝐴𝐵) → 𝐴 ∈ V)
72, 5, 6sylancr 583 . . . . . . . 8 (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐴 ∈ V)
87ad2antrl 721 . . . . . . 7 ((𝑅 We 𝐴 ∧ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝐴 ∈ V)
9 exse 5307 . . . . . . 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 509 . . . . 5 ((𝑅 We 𝐴 ∧ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → (𝑅 We 𝐴𝑅 Se 𝐴))
12 weisoeq 6861 . . . . 5 (((𝑅 We 𝐴𝑅 Se 𝐴) ∧ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝑓 = 𝑔)
1311, 12sylancom 584 . . . 4 ((𝑅 We 𝐴 ∧ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝑓 = 𝑔)
1413ex 403 . . 3 (𝑅 We 𝐴 → ((𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵)) → 𝑓 = 𝑔))
1514alrimivv 2029 . 2 (𝑅 We 𝐴 → ∀𝑓𝑔((𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵)) → 𝑓 = 𝑔))
16 isoeq1 6823 . . 3 (𝑓 = 𝑔 → (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵)))
1716mo4 2639 . 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 226 1 (𝑅 We 𝐴 → ∃*𝑓 𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  wal 1656  wcel 2166  ∃*wmo 2604  Vcvv 3415   Se wse 5300   We wwe 5301  wf 6120  1-1-ontowf1o 6123   Isom wiso 6125
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-reu 3125  df-rmo 3126  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-br 4875  df-opab 4937  df-mpt 4954  df-id 5251  df-po 5264  df-so 5265  df-fr 5302  df-se 5303  df-we 5304  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-isom 6133
This theorem is referenced by:  fzisoeu  40313
  Copyright terms: Public domain W3C validator