Theorem iordsmo 8289

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 6621	. . 3 ⊢ ( I ↾ 𝐴) Fn 𝐴
2		rnresi 6034	. . . 4 ⊢ ran ( I ↾ 𝐴) = 𝐴
3		iordsmo.1	. . . . 5 ⊢ Ord 𝐴
4		ordsson 7728	. . . . 5 ⊢ (Ord 𝐴 → 𝐴 ⊆ On)
5	3, 4	ax-mp 5	. . . 4 ⊢ 𝐴 ⊆ On
6	2, 5	eqsstri 3980	. . 3 ⊢ ran ( I ↾ 𝐴) ⊆ On
7		df-f 6496	. . 3 ⊢ (( I ↾ 𝐴):𝐴⟶On ↔ (( I ↾ 𝐴) Fn 𝐴 ∧ ran ( I ↾ 𝐴) ⊆ On))
8	1, 6, 7	mpbir2an 711	. 2 ⊢ ( I ↾ 𝐴):𝐴⟶On
9		fvresi 7119	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
10	9	adantr 480	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (( I ↾ 𝐴)‘𝑥) = 𝑥)
11		fvresi 7119	. . . . 5 ⊢ (𝑦 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑦) = 𝑦)
12	11	adantl 481	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (( I ↾ 𝐴)‘𝑦) = 𝑦)
13	10, 12	eleq12d 2830	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((( I ↾ 𝐴)‘𝑥) ∈ (( I ↾ 𝐴)‘𝑦) ↔ 𝑥 ∈ 𝑦))
14	13	biimprd 248	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ 𝑦 → (( I ↾ 𝐴)‘𝑥) ∈ (( I ↾ 𝐴)‘𝑦)))
15		dmresi 6011	. 2 ⊢ dom ( I ↾ 𝐴) = 𝐴
16	8, 3, 14, 15	issmo 8280	1 ⊢ Smo ( I ↾ 𝐴)

Syntax hints:   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ⊆ wss 3901   I cid 5518  ran crn 5625   ↾ cres 5626  Ord word 6316  Oncon0 6317   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  Smo wsmo 8277

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-smo 8278

This theorem is referenced by:  smo0  8290