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Theorem xrnidresex 37935
Description: Sufficient condition for a range Cartesian product with restricted identity to be a set. (Contributed by Peter Mazsa, 31-Dec-2021.)
Assertion
Ref Expression
xrnidresex ((𝐴𝑉𝑅𝑊) → (𝑅 ⋉ ( I ↾ 𝐴)) ∈ V)

Proof of Theorem xrnidresex
StepHypRef Expression
1 resiexg 7918 . . 3 (𝐴𝑉 → ( I ↾ 𝐴) ∈ V)
21adantr 479 . 2 ((𝐴𝑉𝑅𝑊) → ( I ↾ 𝐴) ∈ V)
3 xrnresex 37934 . 2 ((𝐴𝑉𝑅𝑊 ∧ ( I ↾ 𝐴) ∈ V) → (𝑅 ⋉ ( I ↾ 𝐴)) ∈ V)
42, 3mpd3an3 1458 1 ((𝐴𝑉𝑅𝑊) → (𝑅 ⋉ ( I ↾ 𝐴)) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wcel 2098  Vcvv 3463   I cid 5569  cres 5674  cxrn 37704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-1st 7991  df-2nd 7992  df-xrn 37899
This theorem is referenced by: (None)
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