Theorem resiexg 5058

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 5041	. . 3 ⊢ Rel ( I ↾ 𝐴)
2		simpr 110	. . . . 5 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
3		eleq1 2294	. . . . . 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 2805	. . . . . 6 ⊢ 𝑦 ∈ V
7	6	opelres 5018	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝐴) ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ 𝐴))
8		df-br 4089	. . . . . . 7 ⊢ (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
9	6	ideq 4882	. . . . . . 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 4755	. . . 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 4817	. 2 ⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
16		xpexg 4840	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → (𝐴 × 𝐴) ∈ V)
17	16	anidms 397	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × 𝐴) ∈ V)
18		ssexg 4228	. 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 2202  Vcvv 2802   ⊆ wss 3200  ⟨cop 3672   class class class wbr 4088   I cid 4385   × cxp 4723   ↾ cres 4727

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-res 4737

This theorem is referenced by:  ordiso  7234  omct  7315  ctssexmid  7348  ssomct  13065  ndxarg  13104  ausgrusgrben  16018  subctctexmid  16601