Theorem smoiso2 8390

Description: The strictly monotone ordinal functions are also isomorphisms of subclasses of On equipped with the membership relation. (Contributed by Mario Carneiro, 20-Mar-2013.)

Assertion

Ref	Expression
smoiso2	⊢ ((Ord 𝐴 ∧ 𝐵 ⊆ On) → ((𝐹:𝐴–onto→𝐵 ∧ Smo 𝐹) ↔ 𝐹 Isom E , E (𝐴, 𝐵)))

Proof of Theorem smoiso2

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fof 6810	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
2		smo11 8385	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ Smo 𝐹) → 𝐹:𝐴–1-1→𝐵)
3	1, 2	sylan 578	. . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Smo 𝐹) → 𝐹:𝐴–1-1→𝐵)
4		simpl 481	. . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Smo 𝐹) → 𝐹:𝐴–onto→𝐵)
5		df-f1o 6556	. . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵))
6	3, 4, 5	sylanbrc 581	. . . . 5 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Smo 𝐹) → 𝐹:𝐴–1-1-onto→𝐵)
7	6	adantl 480	. . . 4 ⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ On) ∧ (𝐹:𝐴–onto→𝐵 ∧ Smo 𝐹)) → 𝐹:𝐴–1-1-onto→𝐵)
8		fofn 6812	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴)
9		smoord 8386	. . . . . . . 8 ⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 ∈ 𝑦 ↔ (𝐹‘𝑥) ∈ (𝐹‘𝑦)))
10		epel 5585	. . . . . . . 8 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
11		fvex 6909	. . . . . . . . 9 ⊢ (𝐹‘𝑦) ∈ V
12	11	epeli 5584	. . . . . . . 8 ⊢ ((𝐹‘𝑥) E (𝐹‘𝑦) ↔ (𝐹‘𝑥) ∈ (𝐹‘𝑦))
13	9, 10, 12	3bitr4g 313	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 E 𝑦 ↔ (𝐹‘𝑥) E (𝐹‘𝑦)))
14	13	ralrimivva 3190	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 E 𝑦 ↔ (𝐹‘𝑥) E (𝐹‘𝑦)))
15	8, 14	sylan 578	. . . . 5 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Smo 𝐹) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 E 𝑦 ↔ (𝐹‘𝑥) E (𝐹‘𝑦)))
16	15	adantl 480	. . . 4 ⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ On) ∧ (𝐹:𝐴–onto→𝐵 ∧ Smo 𝐹)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 E 𝑦 ↔ (𝐹‘𝑥) E (𝐹‘𝑦)))
17		df-isom 6558	. . . 4 ⊢ (𝐹 Isom E , E (𝐴, 𝐵) ↔ (𝐹:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 E 𝑦 ↔ (𝐹‘𝑥) E (𝐹‘𝑦))))
18	7, 16, 17	sylanbrc 581	. . 3 ⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ On) ∧ (𝐹:𝐴–onto→𝐵 ∧ Smo 𝐹)) → 𝐹 Isom E , E (𝐴, 𝐵))
19	18	ex 411	. 2 ⊢ ((Ord 𝐴 ∧ 𝐵 ⊆ On) → ((𝐹:𝐴–onto→𝐵 ∧ Smo 𝐹) → 𝐹 Isom E , E (𝐴, 𝐵)))
20		isof1o 7330	. . . . . . 7 ⊢ (𝐹 Isom E , E (𝐴, 𝐵) → 𝐹:𝐴–1-1-onto→𝐵)
21		f1ofo 6845	. . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–onto→𝐵)
22	20, 21	syl 17	. . . . . 6 ⊢ (𝐹 Isom E , E (𝐴, 𝐵) → 𝐹:𝐴–onto→𝐵)
23	22	3ad2ant1 1130	. . . . 5 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ 𝐵 ⊆ On) → 𝐹:𝐴–onto→𝐵)
24		smoiso 8383	. . . . 5 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ 𝐵 ⊆ On) → Smo 𝐹)
25	23, 24	jca 510	. . . 4 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ 𝐵 ⊆ On) → (𝐹:𝐴–onto→𝐵 ∧ Smo 𝐹))
26	25	3expib 1119	. . 3 ⊢ (𝐹 Isom E , E (𝐴, 𝐵) → ((Ord 𝐴 ∧ 𝐵 ⊆ On) → (𝐹:𝐴–onto→𝐵 ∧ Smo 𝐹)))
27	26	com12 32	. 2 ⊢ ((Ord 𝐴 ∧ 𝐵 ⊆ On) → (𝐹 Isom E , E (𝐴, 𝐵) → (𝐹:𝐴–onto→𝐵 ∧ Smo 𝐹)))
28	19, 27	impbid 211	1 ⊢ ((Ord 𝐴 ∧ 𝐵 ⊆ On) → ((𝐹:𝐴–onto→𝐵 ∧ Smo 𝐹) ↔ 𝐹 Isom E , E (𝐴, 𝐵)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   ∈ wcel 2098  ∀wral 3050   ⊆ wss 3944   class class class wbr 5149   E cep 5581  Ord word 6370  Oncon0 6371   Fn wfn 6544  ⟶wf 6545  –1-1→wf1 6546  –onto→wfo 6547  –1-1-onto→wf1o 6548  ‘cfv 6549   Isom wiso 6550  Smo wsmo 8366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-ord 6374  df-on 6375  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-smo 8367

This theorem is referenced by:  oismo  9565  cofsmo  10294