Theorem smoiso2 8200

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 6688	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
2		smo11 8195	. . . . . . 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 483	. . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Smo 𝐹) → 𝐹:𝐴–onto→𝐵)
5		df-f1o 6440	. . . . . 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 482	. . . 4 ⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ On) ∧ (𝐹:𝐴–onto→𝐵 ∧ Smo 𝐹)) → 𝐹:𝐴–1-1-onto→𝐵)
8		fofn 6690	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴)
9		smoord 8196	. . . . . . . 8 ⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 ∈ 𝑦 ↔ (𝐹‘𝑥) ∈ (𝐹‘𝑦)))
10		epel 5498	. . . . . . . 8 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
11		fvex 6787	. . . . . . . . 9 ⊢ (𝐹‘𝑦) ∈ V
12	11	epeli 5497	. . . . . . . 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 3123	. . . . . 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 482	. . . 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 6442	. . . 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 413	. 2 ⊢ ((Ord 𝐴 ∧ 𝐵 ⊆ On) → ((𝐹:𝐴–onto→𝐵 ∧ Smo 𝐹) → 𝐹 Isom E , E (𝐴, 𝐵)))
20		isof1o 7194	. . . . . . 7 ⊢ (𝐹 Isom E , E (𝐴, 𝐵) → 𝐹:𝐴–1-1-onto→𝐵)
21		f1ofo 6723	. . . . . . 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 1132	. . . . 5 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ 𝐵 ⊆ On) → 𝐹:𝐴–onto→𝐵)
24		smoiso 8193	. . . . 5 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ 𝐵 ⊆ On) → Smo 𝐹)
25	23, 24	jca 512	. . . 4 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ 𝐵 ⊆ On) → (𝐹:𝐴–onto→𝐵 ∧ Smo 𝐹))
26	25	3expib 1121	. . 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 396   ∧ w3a 1086   ∈ wcel 2106  ∀wral 3064   ⊆ wss 3887   class class class wbr 5074   E cep 5494  Ord word 6265  Oncon0 6266   Fn wfn 6428  ⟶wf 6429  –1-1→wf1 6430  –onto→wfo 6431  –1-1-onto→wf1o 6432  ‘cfv 6433   Isom wiso 6434  Smo wsmo 8176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-ord 6269  df-on 6270  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-smo 8177

This theorem is referenced by:  oismo  9299  cofsmo  10025