Theorem resiexg 4673

Description: The existence of a restricted identity function, proved without using the Axiom of Replacement. (Contributed by NM, 13-Jan-2007.)

Assertion

Ref	Expression
resiexg	⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) ∈ V)

Proof of Theorem resiexg

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

Step	Hyp	Ref	Expression
1		relres 4657	. . 3 ⊢ Rel ( I ↾ 𝐴)
2		simpr 108	. . . . 5 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
3		eleq1 2141	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
4	3	biimpa 290	. . . . 5 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴)
5	2, 4	jca 300	. . . 4 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴))
6		vex 2604	. . . . . 6 ⊢ 𝑦 ∈ V
7	6	opelres 4635	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝐴) ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ 𝐴))
8		df-br 3786	. . . . . . 7 ⊢ (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
9	6	ideq 4506	. . . . . . 7 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
10	8, 9	bitr3i 184	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦)
11	10	anbi1i 445	. . . . 5 ⊢ ((⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ 𝐴) ↔ (𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴))
12	7, 11	bitri 182	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝐴) ↔ (𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴))
13		opelxp 4392	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴))
14	5, 12, 13	3imtr4i 199	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴))
15	1, 14	relssi 4449	. 2 ⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
16		xpexg 4470	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → (𝐴 × 𝐴) ∈ V)
17	16	anidms 389	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × 𝐴) ∈ V)
18		ssexg 3917	. 2 ⊢ ((( I ↾ 𝐴) ⊆ (𝐴 × 𝐴) ∧ (𝐴 × 𝐴) ∈ V) → ( I ↾ 𝐴) ∈ V)
19	15, 17, 18	sylancr 405	1 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) ∈ V)

Syntax hints:   → wi 4   ∧ wa 102   ∈ wcel 1433  Vcvv 2601   ⊆ wss 2973  ⟨cop 3401   class class class wbr 3785   I cid 4043   × cxp 4361   ↾ cres 4365

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-13 1444  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  ax-un 4188

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-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-id 4048  df-xp 4369  df-rel 4370  df-res 4375

This theorem is referenced by:  ordiso  6447