Theorem iordsmo 6506

Description: The identity relation restricted to the ordinals is a strictly monotone function. (Contributed by Andrew Salmon, 16-Nov-2011.)

Hypothesis

Ref	Expression
iordsmo.1	⊢ Ord 𝐴

Assertion

Ref	Expression
iordsmo	⊢ Smo ( I ↾ 𝐴)

Proof of Theorem iordsmo

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

Step	Hyp	Ref	Expression
1		fnresi 5457	. . 3 ⊢ ( I ↾ 𝐴) Fn 𝐴
2		rnresi 5100	. . . 4 ⊢ ran ( I ↾ 𝐴) = 𝐴
3		iordsmo.1	. . . . 5 ⊢ Ord 𝐴
4		ordsson 4596	. . . . 5 ⊢ (Ord 𝐴 → 𝐴 ⊆ On)
5	3, 4	ax-mp 5	. . . 4 ⊢ 𝐴 ⊆ On
6	2, 5	eqsstri 3260	. . 3 ⊢ ran ( I ↾ 𝐴) ⊆ On
7		df-f 5337	. . 3 ⊢ (( I ↾ 𝐴):𝐴⟶On ↔ (( I ↾ 𝐴) Fn 𝐴 ∧ ran ( I ↾ 𝐴) ⊆ On))
8	1, 6, 7	mpbir2an 951	. 2 ⊢ ( I ↾ 𝐴):𝐴⟶On
9		fvresi 5855	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
10	9	adantr 276	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (( I ↾ 𝐴)‘𝑥) = 𝑥)
11		fvresi 5855	. . . . 5 ⊢ (𝑦 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑦) = 𝑦)
12	11	adantl 277	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (( I ↾ 𝐴)‘𝑦) = 𝑦)
13	10, 12	eleq12d 2302	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((( I ↾ 𝐴)‘𝑥) ∈ (( I ↾ 𝐴)‘𝑦) ↔ 𝑥 ∈ 𝑦))
14	13	biimprd 158	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ 𝑦 → (( I ↾ 𝐴)‘𝑥) ∈ (( I ↾ 𝐴)‘𝑦)))
15		dmresi 5074	. 2 ⊢ dom ( I ↾ 𝐴) = 𝐴
16	8, 3, 14, 15	issmo 6497	1 ⊢ Smo ( I ↾ 𝐴)

Syntax hints:   ∧ wa 104   = wceq 1398   ∈ wcel 2202   ⊆ wss 3201   I cid 4391  Ord word 4465  Oncon0 4466  ran crn 4732   ↾ cres 4733   Fn wfn 5328  ⟶wf 5329  ‘cfv 5333  Smo wsmo 6494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-smo 6495

This theorem is referenced by:  smo0  6507