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Theorem xrnresex 34479
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 34477 . . 3 ((𝑅𝑆) ↾ 𝐴) = ((𝑅𝐴) ⋉ (𝑆𝐴))
2 xrnres2 34476 . . 3 ((𝑅𝑆) ↾ 𝐴) = (𝑅 ⋉ (𝑆𝐴))
31, 2eqtr3i 2776 . 2 ((𝑅𝐴) ⋉ (𝑆𝐴)) = (𝑅 ⋉ (𝑆𝐴))
4 dfres4 34377 . . . 4 (𝑅𝐴) = (𝑅 ∩ (𝐴 × ran (𝑅𝐴)))
5 dfres4 34377 . . . 4 (𝑆𝐴) = (𝑆 ∩ (𝐴 × ran (𝑆𝐴)))
64, 5xrneq12i 34461 . . 3 ((𝑅𝐴) ⋉ (𝑆𝐴)) = ((𝑅 ∩ (𝐴 × ran (𝑅𝐴))) ⋉ (𝑆 ∩ (𝐴 × ran (𝑆𝐴))))
7 simp1 1130 . . . 4 ((𝐴𝑉𝑅𝑊 ∧ (𝑆𝐴) ∈ 𝑋) → 𝐴𝑉)
8 resexg 5592 . . . . . 6 (𝑅𝑊 → (𝑅𝐴) ∈ V)
9 rnexg 7255 . . . . . 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 7255 . . . . 5 ((𝑆𝐴) ∈ 𝑋 → ran (𝑆𝐴) ∈ V)
13123ad2ant3 1129 . . . 4 ((𝐴𝑉𝑅𝑊 ∧ (𝑆𝐴) ∈ 𝑋) → ran (𝑆𝐴) ∈ V)
14 inxpxrn 34468 . . . . 5 ((𝑅 ∩ (𝐴 × ran (𝑅𝐴))) ⋉ (𝑆 ∩ (𝐴 × ran (𝑆𝐴)))) = ((𝑅𝑆) ∩ (𝐴 × (ran (𝑅𝐴) × ran (𝑆𝐴))))
15 xrninxpex 34467 . . . . 5 ((𝐴𝑉 ∧ ran (𝑅𝐴) ∈ V ∧ ran (𝑆𝐴) ∈ V) → ((𝑅𝑆) ∩ (𝐴 × (ran (𝑅𝐴) × ran (𝑆𝐴)))) ∈ V)
1614, 15syl5eqel 2835 . . . 4 ((𝐴𝑉 ∧ ran (𝑅𝐴) ∈ V ∧ ran (𝑆𝐴) ∈ V) → ((𝑅 ∩ (𝐴 × ran (𝑅𝐴))) ⋉ (𝑆 ∩ (𝐴 × ran (𝑆𝐴)))) ∈ V)
177, 11, 13, 16syl3anc 1473 . . 3 ((𝐴𝑉𝑅𝑊 ∧ (𝑆𝐴) ∈ 𝑋) → ((𝑅 ∩ (𝐴 × ran (𝑅𝐴))) ⋉ (𝑆 ∩ (𝐴 × ran (𝑆𝐴)))) ∈ V)
186, 17syl5eqel 2835 . 2 ((𝐴𝑉𝑅𝑊 ∧ (𝑆𝐴) ∈ 𝑋) → ((𝑅𝐴) ⋉ (𝑆𝐴)) ∈ V)
193, 18syl5eqelr 2836 1 ((𝐴𝑉𝑅𝑊 ∧ (𝑆𝐴) ∈ 𝑋) → (𝑅 ⋉ (𝑆𝐴)) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1072  wcel 2131  Vcvv 3332  cin 3706   × cxp 5256  ran crn 5259  cres 5260  cxrn 34287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-fo 6047  df-fv 6049  df-1st 7325  df-2nd 7326  df-xrn 34448
This theorem is referenced by:  xrnidresex  34480  xrncnvepresex  34481
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