Theorem resiexg 4954

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 4937	. . 3 ⊢ Rel ( I ↾ 𝐴)
2		simpr 110	. . . . 5 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
3		eleq1 2240	. . . . . 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 2742	. . . . . 6 ⊢ 𝑦 ∈ V
7	6	opelres 4914	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝐴) ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ 𝐴))
8		df-br 4006	. . . . . . 7 ⊢ (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
9	6	ideq 4781	. . . . . . 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 4658	. . . 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 4719	. 2 ⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
16		xpexg 4742	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → (𝐴 × 𝐴) ∈ V)
17	16	anidms 397	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × 𝐴) ∈ V)
18		ssexg 4144	. 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 2148  Vcvv 2739   ⊆ wss 3131  ⟨cop 3597   class class class wbr 4005   I cid 4290   × cxp 4626   ↾ cres 4630

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-res 4640

This theorem is referenced by:  ordiso  7037  omct  7118  ctssexmid  7150  ssomct  12448  ndxarg  12487  subctctexmid  14835