Theorem smoiso2 8338

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 6772	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
2		smo11 8333	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ Smo 𝐹) → 𝐹:𝐴–1-1→𝐵)
3	1, 2	sylan 580	. . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Smo 𝐹) → 𝐹:𝐴–1-1→𝐵)
4		simpl 482	. . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Smo 𝐹) → 𝐹:𝐴–onto→𝐵)
5		df-f1o 6518	. . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵))
6	3, 4, 5	sylanbrc 583	. . . . 5 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Smo 𝐹) → 𝐹:𝐴–1-1-onto→𝐵)
7	6	adantl 481	. . . 4 ⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ On) ∧ (𝐹:𝐴–onto→𝐵 ∧ Smo 𝐹)) → 𝐹:𝐴–1-1-onto→𝐵)
8		fofn 6774	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴)
9		smoord 8334	. . . . . . . 8 ⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 ∈ 𝑦 ↔ (𝐹‘𝑥) ∈ (𝐹‘𝑦)))
10		epel 5541	. . . . . . . 8 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
11		fvex 6871	. . . . . . . . 9 ⊢ (𝐹‘𝑦) ∈ V
12	11	epeli 5540	. . . . . . . 8 ⊢ ((𝐹‘𝑥) E (𝐹‘𝑦) ↔ (𝐹‘𝑥) ∈ (𝐹‘𝑦))
13	9, 10, 12	3bitr4g 314	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 E 𝑦 ↔ (𝐹‘𝑥) E (𝐹‘𝑦)))
14	13	ralrimivva 3180	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 E 𝑦 ↔ (𝐹‘𝑥) E (𝐹‘𝑦)))
15	8, 14	sylan 580	. . . . 5 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Smo 𝐹) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 E 𝑦 ↔ (𝐹‘𝑥) E (𝐹‘𝑦)))
16	15	adantl 481	. . . 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 6520	. . . 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 583	. . 3 ⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ On) ∧ (𝐹:𝐴–onto→𝐵 ∧ Smo 𝐹)) → 𝐹 Isom E , E (𝐴, 𝐵))
19	18	ex 412	. 2 ⊢ ((Ord 𝐴 ∧ 𝐵 ⊆ On) → ((𝐹:𝐴–onto→𝐵 ∧ Smo 𝐹) → 𝐹 Isom E , E (𝐴, 𝐵)))
20		isof1o 7298	. . . . . . 7 ⊢ (𝐹 Isom E , E (𝐴, 𝐵) → 𝐹:𝐴–1-1-onto→𝐵)
21		f1ofo 6807	. . . . . . 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 1133	. . . . 5 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ 𝐵 ⊆ On) → 𝐹:𝐴–onto→𝐵)
24		smoiso 8331	. . . . 5 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ 𝐵 ⊆ On) → Smo 𝐹)
25	23, 24	jca 511	. . . 4 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ 𝐵 ⊆ On) → (𝐹:𝐴–onto→𝐵 ∧ Smo 𝐹))
26	25	3expib 1122	. . 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 212	1 ⊢ ((Ord 𝐴 ∧ 𝐵 ⊆ On) → ((𝐹:𝐴–onto→𝐵 ∧ Smo 𝐹) ↔ 𝐹 Isom E , E (𝐴, 𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2109  ∀wral 3044   ⊆ wss 3914   class class class wbr 5107   E cep 5537  Ord word 6331  Oncon0 6332   Fn wfn 6506  ⟶wf 6507  –1-1→wf1 6508  –onto→wfo 6509  –1-1-onto→wf1o 6510  ‘cfv 6511   Isom wiso 6512  Smo wsmo 8314

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-ord 6335  df-on 6336  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-isom 6520  df-smo 8315

This theorem is referenced by:  oismo  9493  cofsmo  10222