Theorem iordsmo 8350

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 6668	. . 3 ⊢ ( I ↾ 𝐴) Fn 𝐴
2		rnresi 6079	. . . 4 ⊢ ran ( I ↾ 𝐴) = 𝐴
3		iordsmo.1	. . . . 5 ⊢ Ord 𝐴
4		ordsson 7788	. . . . 5 ⊢ (Ord 𝐴 → 𝐴 ⊆ On)
5	3, 4	ax-mp 5	. . . 4 ⊢ 𝐴 ⊆ On
6	2, 5	eqsstri 3984	. . 3 ⊢ ran ( I ↾ 𝐴) ⊆ On
7		df-f 6544	. . 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 7177	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
10	9	adantr 486	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (( I ↾ 𝐴)‘𝑥) = 𝑥)
11		fvresi 7177	. . . . 5 ⊢ (𝑦 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑦) = 𝑦)
12	11	adantl 487	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (( I ↾ 𝐴)‘𝑦) = 𝑦)
13	10, 12	eleq12d 2859	. . 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 6056	. 2 ⊢ dom ( I ↾ 𝐴) = 𝐴
16	8, 3, 14, 15	issmo 8341	1 ⊢ Smo ( I ↾ 𝐴)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ⊆ wss 3906   I cid 5557  ran crn 5664   ↾ cres 5665  Ord word 6363  Oncon0 6364   Fn wfn 6535  ⟶wf 6536  ‘cfv 6540  Smo wsmo 8338

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406

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 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-ord 6367  df-on 6368  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-smo 8339

This theorem is used by:  smo0  8351