Theorem iordsmo 8403

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 6698	. . 3 ⊢ ( I ↾ 𝐴) Fn 𝐴
2		rnresi 6091	. . . 4 ⊢ ran ( I ↾ 𝐴) = 𝐴
3		iordsmo.1	. . . . 5 ⊢ Ord 𝐴
4		ordsson 7831	. . . . 5 ⊢ (Ord 𝐴 → 𝐴 ⊆ On)
5	3, 4	ax-mp 5	. . . 4 ⊢ 𝐴 ⊆ On
6	2, 5	eqsstri 3959	. . 3 ⊢ ran ( I ↾ 𝐴) ⊆ On
7		df-f 6570	. . 3 ⊢ (( I ↾ 𝐴):𝐴⟶On ↔ (( I ↾ 𝐴) Fn 𝐴 ∧ ran ( I ↾ 𝐴) ⊆ On))
8	1, 6, 7	mpbir2an 724	. 2 ⊢ ( I ↾ 𝐴):𝐴⟶On
9		fvresi 7209	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
10	9	adantr 486	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (( I ↾ 𝐴)‘𝑥) = 𝑥)
11		fvresi 7209	. . . . 5 ⊢ (𝑦 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑦) = 𝑦)
12	11	adantl 487	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (( I ↾ 𝐴)‘𝑦) = 𝑦)
13	10, 12	eleq12d 2858	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((( I ↾ 𝐴)‘𝑥) ∈ (( I ↾ 𝐴)‘𝑦) ↔ 𝑥 ∈ 𝑦))
14	13	biimprd 251	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ 𝑦 → (( I ↾ 𝐴)‘𝑥) ∈ (( I ↾ 𝐴)‘𝑦)))
15		dmresi 6062	. 2 ⊢ dom ( I ↾ 𝐴) = 𝐴
16	8, 3, 14, 15	issmo 8394	1 ⊢ Smo ( I ↾ 𝐴)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ⊆ wss 3900   I cid 5552  ran crn 5659   ↾ cres 5660  Ord word 6387  Oncon0 6388   Fn wfn 6561  ⟶wf 6562  ‘cfv 6566  Smo wsmo 8391

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-ord 6391  df-on 6392  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-fv 6574  df-smo 8392

This theorem is used by:  smo0  8404