Theorem wemoiso 7926

Description: Thus, there is at most one isomorphism between any two well-ordered sets. TODO: Shorten finnisoeu 10035. (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 3433	. . . . . . . . 9 ⊢ 𝑓 ∈ V
3		isof1o 7278	. . . . . . . . . 10 ⊢ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝑓:𝐴–1-1-onto→𝐵)
4		f1of 6780	. . . . . . . . . 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 7856	. . . . . . . . 9 ⊢ ((𝑓 ∈ V ∧ 𝑓:𝐴⟶𝐵) → 𝐴 ∈ V)
7	2, 5, 6	sylancr 588	. . . . . . . 8 ⊢ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐴 ∈ V)
8	7	ad2antrl 729	. . . . . . 7 ⊢ ((𝑅 We 𝐴 ∧ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝐴 ∈ V)
9		exse 5591	. . . . . . 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 7310	. . . . 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 589	. . . 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 1930	. 2 ⊢ (𝑅 We 𝐴 → ∀𝑓∀𝑔((𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵)) → 𝑓 = 𝑔))
16		isoeq1 7272	. . 3 ⊢ (𝑓 = 𝑔 → (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵)))
17	16	mo4 2566	. 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 1540   ∈ wcel 2114  ∃*wmo 2537  Vcvv 3429   Se wse 5582   We wwe 5583  ⟶wf 6494  –1-1-onto→wf1o 6497   Isom wiso 6499

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507

This theorem is referenced by:  fzisoeu  45733