Theorem resiexg 4988

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 4971	. . 3 ⊢ Rel ( I ↾ 𝐴)
2		simpr 110	. . . . 5 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
3		eleq1 2256	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
4	3	biimpa 296	. . . . 5 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴)
5	2, 4	jca 306	. . . 4 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴))
6		vex 2763	. . . . . 6 ⊢ 𝑦 ∈ V
7	6	opelres 4948	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝐴) ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ 𝐴))
8		df-br 4031	. . . . . . 7 ⊢ (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
9	6	ideq 4815	. . . . . . 7 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
10	8, 9	bitr3i 186	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦)
11	10	anbi1i 458	. . . . 5 ⊢ ((⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ 𝐴) ↔ (𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴))
12	7, 11	bitri 184	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝐴) ↔ (𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴))
13		opelxp 4690	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴))
14	5, 12, 13	3imtr4i 201	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴))
15	1, 14	relssi 4751	. 2 ⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
16		xpexg 4774	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → (𝐴 × 𝐴) ∈ V)
17	16	anidms 397	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × 𝐴) ∈ V)
18		ssexg 4169	. 2 ⊢ ((( I ↾ 𝐴) ⊆ (𝐴 × 𝐴) ∧ (𝐴 × 𝐴) ∈ V) → ( I ↾ 𝐴) ∈ V)
19	15, 17, 18	sylancr 414	1 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) ∈ V)

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2164  Vcvv 2760   ⊆ wss 3154  ⟨cop 3622   class class class wbr 4030   I cid 4320   × cxp 4658   ↾ cres 4662

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-id 4325  df-xp 4666  df-rel 4667  df-res 4672

This theorem is referenced by:  ordiso  7097  omct  7178  ctssexmid  7211  ssomct  12605  ndxarg  12644  subctctexmid  15561