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Theorem xrnresex 35521
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 35519 . . 3 ((𝑅𝑆) ↾ 𝐴) = ((𝑅𝐴) ⋉ (𝑆𝐴))
2 xrnres2 35518 . . 3 ((𝑅𝑆) ↾ 𝐴) = (𝑅 ⋉ (𝑆𝐴))
31, 2eqtr3i 2850 . 2 ((𝑅𝐴) ⋉ (𝑆𝐴)) = (𝑅 ⋉ (𝑆𝐴))
4 dfres4 35418 . . . 4 (𝑅𝐴) = (𝑅 ∩ (𝐴 × ran (𝑅𝐴)))
5 dfres4 35418 . . . 4 (𝑆𝐴) = (𝑆 ∩ (𝐴 × ran (𝑆𝐴)))
64, 5xrneq12i 35503 . . 3 ((𝑅𝐴) ⋉ (𝑆𝐴)) = ((𝑅 ∩ (𝐴 × ran (𝑅𝐴))) ⋉ (𝑆 ∩ (𝐴 × ran (𝑆𝐴))))
7 simp1 1130 . . . 4 ((𝐴𝑉𝑅𝑊 ∧ (𝑆𝐴) ∈ 𝑋) → 𝐴𝑉)
8 resexg 5896 . . . . . 6 (𝑅𝑊 → (𝑅𝐴) ∈ V)
9 rnexg 7605 . . . . . 6 ((𝑅𝐴) ∈ V → ran (𝑅𝐴) ∈ V)
108, 9syl 17 . . . . 5 (𝑅𝑊 → ran (𝑅𝐴) ∈ V)
11103ad2ant2 1128 . . . 4 ((𝐴𝑉𝑅𝑊 ∧ (𝑆𝐴) ∈ 𝑋) → ran (𝑅𝐴) ∈ V)
12 rnexg 7605 . . . . 5 ((𝑆𝐴) ∈ 𝑋 → ran (𝑆𝐴) ∈ V)
13123ad2ant3 1129 . . . 4 ((𝐴𝑉𝑅𝑊 ∧ (𝑆𝐴) ∈ 𝑋) → ran (𝑆𝐴) ∈ V)
14 inxpxrn 35510 . . . . 5 ((𝑅 ∩ (𝐴 × ran (𝑅𝐴))) ⋉ (𝑆 ∩ (𝐴 × ran (𝑆𝐴)))) = ((𝑅𝑆) ∩ (𝐴 × (ran (𝑅𝐴) × ran (𝑆𝐴))))
15 xrninxpex 35509 . . . . 5 ((𝐴𝑉 ∧ ran (𝑅𝐴) ∈ V ∧ ran (𝑆𝐴) ∈ V) → ((𝑅𝑆) ∩ (𝐴 × (ran (𝑅𝐴) × ran (𝑆𝐴)))) ∈ V)
1614, 15eqeltrid 2921 . . . 4 ((𝐴𝑉 ∧ ran (𝑅𝐴) ∈ V ∧ ran (𝑆𝐴) ∈ V) → ((𝑅 ∩ (𝐴 × ran (𝑅𝐴))) ⋉ (𝑆 ∩ (𝐴 × ran (𝑆𝐴)))) ∈ V)
177, 11, 13, 16syl3anc 1365 . . 3 ((𝐴𝑉𝑅𝑊 ∧ (𝑆𝐴) ∈ 𝑋) → ((𝑅 ∩ (𝐴 × ran (𝑅𝐴))) ⋉ (𝑆 ∩ (𝐴 × ran (𝑆𝐴)))) ∈ V)
186, 17eqeltrid 2921 . 2 ((𝐴𝑉𝑅𝑊 ∧ (𝑆𝐴) ∈ 𝑋) → ((𝑅𝐴) ⋉ (𝑆𝐴)) ∈ V)
193, 18eqeltrrid 2922 1 ((𝐴𝑉𝑅𝑊 ∧ (𝑆𝐴) ∈ 𝑋) → (𝑅 ⋉ (𝑆𝐴)) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1081  wcel 2107  Vcvv 3499  cin 3938   × cxp 5551  ran crn 5554  cres 5555  cxrn 35320
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-fo 6357  df-fv 6359  df-1st 7683  df-2nd 7684  df-xrn 35490
This theorem is referenced by:  xrnidresex  35522  xrncnvepresex  35523
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