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Theorem resiexg 4681
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 (𝐴𝑉 → ( I ↾ 𝐴) ∈ V)

Proof of Theorem resiexg
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relres 4667 . . 3 Rel ( I ↾ 𝐴)
2 simpr 107 . . . . 5 ((𝑥 = 𝑦𝑥𝐴) → 𝑥𝐴)
3 eleq1 2116 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
43biimpa 284 . . . . 5 ((𝑥 = 𝑦𝑥𝐴) → 𝑦𝐴)
52, 4jca 294 . . . 4 ((𝑥 = 𝑦𝑥𝐴) → (𝑥𝐴𝑦𝐴))
6 vex 2577 . . . . . 6 𝑦 ∈ V
76opelres 4645 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝐴) ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥𝐴))
8 df-br 3793 . . . . . . 7 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
96ideq 4516 . . . . . . 7 (𝑥 I 𝑦𝑥 = 𝑦)
108, 9bitr3i 179 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦)
1110anbi1i 439 . . . . 5 ((⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥𝐴) ↔ (𝑥 = 𝑦𝑥𝐴))
127, 11bitri 177 . . . 4 (⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝐴) ↔ (𝑥 = 𝑦𝑥𝐴))
13 opelxp 4402 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴) ↔ (𝑥𝐴𝑦𝐴))
145, 12, 133imtr4i 194 . . 3 (⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴))
151, 14relssi 4459 . 2 ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
16 xpexg 4480 . . 3 ((𝐴𝑉𝐴𝑉) → (𝐴 × 𝐴) ∈ V)
1716anidms 383 . 2 (𝐴𝑉 → (𝐴 × 𝐴) ∈ V)
18 ssexg 3924 . 2 ((( I ↾ 𝐴) ⊆ (𝐴 × 𝐴) ∧ (𝐴 × 𝐴) ∈ V) → ( I ↾ 𝐴) ∈ V)
1915, 17, 18sylancr 399 1 (𝐴𝑉 → ( I ↾ 𝐴) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wcel 1409  Vcvv 2574  wss 2945  cop 3406   class class class wbr 3792   I cid 4053   × cxp 4371  cres 4375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-res 4385
This theorem is referenced by:  ordiso  6416
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