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Theorem xrnresex 36095
 Description: Sufficient condition for a restricted range Cartesian product to be a set. (Contributed by Peter Mazsa, 16-Dec-2020.) (Revised by Peter Mazsa, 7-Sep-2021.)
Assertion
Ref Expression
xrnresex ((𝐴𝑉𝑅𝑊 ∧ (𝑆𝐴) ∈ 𝑋) → (𝑅 ⋉ (𝑆𝐴)) ∈ V)

Proof of Theorem xrnresex
StepHypRef Expression
1 xrnres3 36093 . . 3 ((𝑅𝑆) ↾ 𝐴) = ((𝑅𝐴) ⋉ (𝑆𝐴))
2 xrnres2 36092 . . 3 ((𝑅𝑆) ↾ 𝐴) = (𝑅 ⋉ (𝑆𝐴))
31, 2eqtr3i 2784 . 2 ((𝑅𝐴) ⋉ (𝑆𝐴)) = (𝑅 ⋉ (𝑆𝐴))
4 dfres4 35991 . . . 4 (𝑅𝐴) = (𝑅 ∩ (𝐴 × ran (𝑅𝐴)))
5 dfres4 35991 . . . 4 (𝑆𝐴) = (𝑆 ∩ (𝐴 × ran (𝑆𝐴)))
64, 5xrneq12i 36077 . . 3 ((𝑅𝐴) ⋉ (𝑆𝐴)) = ((𝑅 ∩ (𝐴 × ran (𝑅𝐴))) ⋉ (𝑆 ∩ (𝐴 × ran (𝑆𝐴))))
7 simp1 1134 . . . 4 ((𝐴𝑉𝑅𝑊 ∧ (𝑆𝐴) ∈ 𝑋) → 𝐴𝑉)
8 resexg 5870 . . . . . 6 (𝑅𝑊 → (𝑅𝐴) ∈ V)
9 rnexg 7615 . . . . . 6 ((𝑅𝐴) ∈ V → ran (𝑅𝐴) ∈ V)
108, 9syl 17 . . . . 5 (𝑅𝑊 → ran (𝑅𝐴) ∈ V)
11103ad2ant2 1132 . . . 4 ((𝐴𝑉𝑅𝑊 ∧ (𝑆𝐴) ∈ 𝑋) → ran (𝑅𝐴) ∈ V)
12 rnexg 7615 . . . . 5 ((𝑆𝐴) ∈ 𝑋 → ran (𝑆𝐴) ∈ V)
13123ad2ant3 1133 . . . 4 ((𝐴𝑉𝑅𝑊 ∧ (𝑆𝐴) ∈ 𝑋) → ran (𝑆𝐴) ∈ V)
14 inxpxrn 36084 . . . . 5 ((𝑅 ∩ (𝐴 × ran (𝑅𝐴))) ⋉ (𝑆 ∩ (𝐴 × ran (𝑆𝐴)))) = ((𝑅𝑆) ∩ (𝐴 × (ran (𝑅𝐴) × ran (𝑆𝐴))))
15 xrninxpex 36083 . . . . 5 ((𝐴𝑉 ∧ ran (𝑅𝐴) ∈ V ∧ ran (𝑆𝐴) ∈ V) → ((𝑅𝑆) ∩ (𝐴 × (ran (𝑅𝐴) × ran (𝑆𝐴)))) ∈ V)
1614, 15eqeltrid 2857 . . . 4 ((𝐴𝑉 ∧ ran (𝑅𝐴) ∈ V ∧ ran (𝑆𝐴) ∈ V) → ((𝑅 ∩ (𝐴 × ran (𝑅𝐴))) ⋉ (𝑆 ∩ (𝐴 × ran (𝑆𝐴)))) ∈ V)
177, 11, 13, 16syl3anc 1369 . . 3 ((𝐴𝑉𝑅𝑊 ∧ (𝑆𝐴) ∈ 𝑋) → ((𝑅 ∩ (𝐴 × ran (𝑅𝐴))) ⋉ (𝑆 ∩ (𝐴 × ran (𝑆𝐴)))) ∈ V)
186, 17eqeltrid 2857 . 2 ((𝐴𝑉𝑅𝑊 ∧ (𝑆𝐴) ∈ 𝑋) → ((𝑅𝐴) ⋉ (𝑆𝐴)) ∈ V)
193, 18eqeltrrid 2858 1 ((𝐴𝑉𝑅𝑊 ∧ (𝑆𝐴) ∈ 𝑋) → (𝑅 ⋉ (𝑆𝐴)) ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1085   ∈ wcel 2112  Vcvv 3410   ∩ cin 3858   × cxp 5523  ran crn 5526   ↾ cres 5527   ⋉ cxrn 35893 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-fo 6342  df-fv 6344  df-1st 7694  df-2nd 7695  df-xrn 36064 This theorem is referenced by:  xrnidresex  36096  xrncnvepresex  36097
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