Theorem wemoiso 7905

Description: Thus, there is at most one isomorphism between any two well-ordered sets. TODO: Shorten finnisoeu 10004. (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.

Step	Hyp	Ref	Expression
1		simpl 482	. . . . . 6 ⊢ ((𝑅 We 𝐴 ∧ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝑅 We 𝐴)
2		vex 3440	. . . . . . . . 9 ⊢ 𝑓 ∈ V
3		isof1o 7257	. . . . . . . . . 10 ⊢ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝑓:𝐴–1-1-onto→𝐵)
4		f1of 6763	. . . . . . . . . 10 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴⟶𝐵)
5	3, 4	syl 17	. . . . . . . . 9 ⊢ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝑓:𝐴⟶𝐵)
6		dmfex 7835	. . . . . . . . 9 ⊢ ((𝑓 ∈ V ∧ 𝑓:𝐴⟶𝐵) → 𝐴 ∈ V)
7	2, 5, 6	sylancr 587	. . . . . . . 8 ⊢ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐴 ∈ V)
8	7	ad2antrl 728	. . . . . . 7 ⊢ ((𝑅 We 𝐴 ∧ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝐴 ∈ V)
9		exse 5574	. . . . . . 7 ⊢ (𝐴 ∈ V → 𝑅 Se 𝐴)
10	8, 9	syl 17	. . . . . 6 ⊢ ((𝑅 We 𝐴 ∧ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝑅 Se 𝐴)
11	1, 10	jca 511	. . . . 5 ⊢ ((𝑅 We 𝐴 ∧ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → (𝑅 We 𝐴 ∧ 𝑅 Se 𝐴))
12		weisoeq 7289	. . . . 5 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝑓 = 𝑔)
13	11, 12	sylancom 588	. . . 4 ⊢ ((𝑅 We 𝐴 ∧ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝑓 = 𝑔)
14	13	ex 412	. . 3 ⊢ (𝑅 We 𝐴 → ((𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵)) → 𝑓 = 𝑔))
15	14	alrimivv 1929	. 2 ⊢ (𝑅 We 𝐴 → ∀𝑓∀𝑔((𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵)) → 𝑓 = 𝑔))
16		isoeq1 7251	. . 3 ⊢ (𝑓 = 𝑔 → (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵)))
17	16	mo4 2561	. 2 ⊢ (∃*𝑓 𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ ∀𝑓∀𝑔((𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵)) → 𝑓 = 𝑔))
18	15, 17	sylibr 234	1 ⊢ (𝑅 We 𝐴 → ∃*𝑓 𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1539   ∈ wcel 2111  ∃*wmo 2533  Vcvv 3436   Se wse 5565   We wwe 5566  ⟶wf 6477  –1-1-onto→wf1o 6480   Isom wiso 6482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-isom 6490

This theorem is referenced by:  fzisoeu  45411