Theorem iordsmo 5935

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 5036	. . 3 ⊢ ( I ↾ 𝐴) Fn 𝐴
2		rnresi 4702	. . . 4 ⊢ ran ( I ↾ 𝐴) = 𝐴
3		iordsmo.1	. . . . 5 ⊢ Ord 𝐴
4		ordsson 4236	. . . . 5 ⊢ (Ord 𝐴 → 𝐴 ⊆ On)
5	3, 4	ax-mp 7	. . . 4 ⊢ 𝐴 ⊆ On
6	2, 5	eqsstri 3029	. . 3 ⊢ ran ( I ↾ 𝐴) ⊆ On
7		df-f 4926	. . 3 ⊢ (( I ↾ 𝐴):𝐴⟶On ↔ (( I ↾ 𝐴) Fn 𝐴 ∧ ran ( I ↾ 𝐴) ⊆ On))
8	1, 6, 7	mpbir2an 883	. 2 ⊢ ( I ↾ 𝐴):𝐴⟶On
9		fvresi 5377	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
10	9	adantr 270	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (( I ↾ 𝐴)‘𝑥) = 𝑥)
11		fvresi 5377	. . . . 5 ⊢ (𝑦 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑦) = 𝑦)
12	11	adantl 271	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (( I ↾ 𝐴)‘𝑦) = 𝑦)
13	10, 12	eleq12d 2149	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((( I ↾ 𝐴)‘𝑥) ∈ (( I ↾ 𝐴)‘𝑦) ↔ 𝑥 ∈ 𝑦))
14	13	biimprd 156	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ 𝑦 → (( I ↾ 𝐴)‘𝑥) ∈ (( I ↾ 𝐴)‘𝑦)))
15		dmresi 4681	. 2 ⊢ dom ( I ↾ 𝐴) = 𝐴
16	8, 3, 14, 15	issmo 5926	1 ⊢ Smo ( I ↾ 𝐴)

Syntax hints:   ∧ wa 102   = wceq 1284   ∈ wcel 1433   ⊆ wss 2973   I cid 4043  Ord word 4117  Oncon0 4118  ran crn 4364   ↾ cres 4365   Fn wfn 4917  ⟶wf 4918  ‘cfv 4922  Smo wsmo 5923

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-fv 4930  df-smo 5924

This theorem is referenced by:  smo0  5936