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Theorem xrncnvepresex 36520
Description: Sufficient condition for a range Cartesian product with restricted converse epsilon to be a set. (Contributed by Peter Mazsa, 16-Dec-2020.) (Revised by Peter Mazsa, 23-Sep-2021.)
Assertion
Ref Expression
xrncnvepresex ((𝐴𝑉𝑅𝑊) → (𝑅 ⋉ ( E ↾ 𝐴)) ∈ V)

Proof of Theorem xrncnvepresex
StepHypRef Expression
1 cnvepresex 36455 . . 3 (𝐴𝑉 → ( E ↾ 𝐴) ∈ V)
21adantr 481 . 2 ((𝐴𝑉𝑅𝑊) → ( E ↾ 𝐴) ∈ V)
3 xrnresex 36518 . 2 ((𝐴𝑉𝑅𝑊 ∧ ( E ↾ 𝐴) ∈ V) → (𝑅 ⋉ ( E ↾ 𝐴)) ∈ V)
42, 3mpd3an3 1461 1 ((𝐴𝑉𝑅𝑊) → (𝑅 ⋉ ( E ↾ 𝐴)) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2106  Vcvv 3430   E cep 5490  ccnv 5584  cres 5587  cxrn 36318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5222  ax-nul 5229  ax-pow 5287  ax-pr 5351  ax-un 7579
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3432  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4258  df-if 4461  df-pw 4536  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-iun 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5485  df-eprel 5491  df-xp 5591  df-rel 5592  df-cnv 5593  df-co 5594  df-dm 5595  df-rn 5596  df-res 5597  df-ima 5598  df-iota 6385  df-fun 6429  df-fn 6430  df-f 6431  df-f1 6432  df-fo 6433  df-f1o 6434  df-fv 6435  df-1st 7821  df-2nd 7822  df-xrn 36487
This theorem is referenced by:  1cossxrncnvepresex  36531
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