Theorem wemoiso2 7972

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.

Step	Hyp	Ref	Expression
1		simpl 487	. . . . . 6 ⊢ ((𝑆 We 𝐵 ∧ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝑆 We 𝐵)
2		isof1o 7323	. . . . . . . . . 10 ⊢ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝑓:𝐴–1-1-onto→𝐵)
3		f1ofo 6830	. . . . . . . . . 10 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴–onto→𝐵)
4		forn 6797	. . . . . . . . . 10 ⊢ (𝑓:𝐴–onto→𝐵 → ran 𝑓 = 𝐵)
5	2, 3, 4	3syl 19	. . . . . . . . 9 ⊢ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ran 𝑓 = 𝐵)
6		vex 3459	. . . . . . . . . 10 ⊢ 𝑓 ∈ V
7	6	rnex 7908	. . . . . . . . 9 ⊢ ran 𝑓 ∈ V
8	5, 7	eqeltrrdi 2872	. . . . . . . 8 ⊢ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐵 ∈ V)
9	8	ad2antrl 740	. . . . . . 7 ⊢ ((𝑆 We 𝐵 ∧ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝐵 ∈ V)
10		exse 5623	. . . . . . 7 ⊢ (𝐵 ∈ V → 𝑆 Se 𝐵)
11	9, 10	syl 18	. . . . . 6 ⊢ ((𝑆 We 𝐵 ∧ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝑆 Se 𝐵)
12	1, 11	jca 520	. . . . 5 ⊢ ((𝑆 We 𝐵 ∧ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → (𝑆 We 𝐵 ∧ 𝑆 Se 𝐵))
13		weisoeq2 7356	. . . . 5 ⊢ (((𝑆 We 𝐵 ∧ 𝑆 Se 𝐵) ∧ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝑓 = 𝑔)
14	12, 13	sylancom 599	. . . 4 ⊢ ((𝑆 We 𝐵 ∧ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝑓 = 𝑔)
15	14	ex 417	. . 3 ⊢ (𝑆 We 𝐵 → ((𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵)) → 𝑓 = 𝑔))
16	15	alrimivv 1958	. 2 ⊢ (𝑆 We 𝐵 → ∀𝑓∀𝑔((𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵)) → 𝑓 = 𝑔))
17		isoeq1 7317	. . 3 ⊢ (𝑓 = 𝑔 → (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵)))
18	17	mo4 2594	. 2 ⊢ (∃*𝑓 𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ ∀𝑓∀𝑔((𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵)) → 𝑓 = 𝑔))
19	16, 18	sylibr 237	1 ⊢ (𝑆 We 𝐵 → ∃*𝑓 𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵))

Syntax hints:   → wi 4   ∧ wa 400  ∀wal 1568   = wceq 1570   ∈ wcel 2143  ∃*wmo 2565  Vcvv 3455   Se wse 5614   We wwe 5615  ran crn 5664  –onto→wfo 6536  –1-1-onto→wf1o 6537   Isom wiso 6539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547

This theorem is referenced by:  finnisoeu  10098