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Theorem xrnidresex 35693
 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 7594 . . 3 (𝐴𝑉 → ( I ↾ 𝐴) ∈ V)
21adantr 484 . 2 ((𝐴𝑉𝑅𝑊) → ( I ↾ 𝐴) ∈ V)
3 xrnresex 35692 . 2 ((𝐴𝑉𝑅𝑊 ∧ ( I ↾ 𝐴) ∈ V) → (𝑅 ⋉ ( I ↾ 𝐴)) ∈ V)
42, 3mpd3an3 1459 1 ((𝐴𝑉𝑅𝑊) → (𝑅 ⋉ ( I ↾ 𝐴)) ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2115  Vcvv 3471   I cid 5432   ↾ cres 5530   ⋉ cxrn 35490 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fo 6334  df-fv 6336  df-1st 7664  df-2nd 7665  df-xrn 35661 This theorem is referenced by: (None)
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