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Theorem resiexg 4952
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 4935 . . 3 Rel ( I ↾ 𝐴)
2 simpr 110 . . . . 5 ((𝑥 = 𝑦𝑥𝐴) → 𝑥𝐴)
3 eleq1 2240 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
43biimpa 296 . . . . 5 ((𝑥 = 𝑦𝑥𝐴) → 𝑦𝐴)
52, 4jca 306 . . . 4 ((𝑥 = 𝑦𝑥𝐴) → (𝑥𝐴𝑦𝐴))
6 vex 2740 . . . . . 6 𝑦 ∈ V
76opelres 4912 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝐴) ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥𝐴))
8 df-br 4004 . . . . . . 7 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
96ideq 4779 . . . . . . 7 (𝑥 I 𝑦𝑥 = 𝑦)
108, 9bitr3i 186 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦)
1110anbi1i 458 . . . . 5 ((⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥𝐴) ↔ (𝑥 = 𝑦𝑥𝐴))
127, 11bitri 184 . . . 4 (⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝐴) ↔ (𝑥 = 𝑦𝑥𝐴))
13 opelxp 4656 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴) ↔ (𝑥𝐴𝑦𝐴))
145, 12, 133imtr4i 201 . . 3 (⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴))
151, 14relssi 4717 . 2 ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
16 xpexg 4740 . . 3 ((𝐴𝑉𝐴𝑉) → (𝐴 × 𝐴) ∈ V)
1716anidms 397 . 2 (𝐴𝑉 → (𝐴 × 𝐴) ∈ V)
18 ssexg 4142 . 2 ((( I ↾ 𝐴) ⊆ (𝐴 × 𝐴) ∧ (𝐴 × 𝐴) ∈ V) → ( I ↾ 𝐴) ∈ V)
1915, 17, 18sylancr 414 1 (𝐴𝑉 → ( I ↾ 𝐴) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wcel 2148  Vcvv 2737  wss 3129  cop 3595   class class class wbr 4003   I cid 4288   × cxp 4624  cres 4628
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 4121  ax-pow 4174  ax-pr 4209  ax-un 4433
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 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-res 4638
This theorem is referenced by:  ordiso  7034  omct  7115  ctssexmid  7147  ssomct  12445  ndxarg  12484  subctctexmid  14686
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