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Theorem xrnresex 37910
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 37908 . . 3 ((𝑅𝑆) ↾ 𝐴) = ((𝑅𝐴) ⋉ (𝑆𝐴))
2 xrnres2 37907 . . 3 ((𝑅𝑆) ↾ 𝐴) = (𝑅 ⋉ (𝑆𝐴))
31, 2eqtr3i 2758 . 2 ((𝑅𝐴) ⋉ (𝑆𝐴)) = (𝑅 ⋉ (𝑆𝐴))
4 dfres4 37797 . . . 4 (𝑅𝐴) = (𝑅 ∩ (𝐴 × ran (𝑅𝐴)))
5 dfres4 37797 . . . 4 (𝑆𝐴) = (𝑆 ∩ (𝐴 × ran (𝑆𝐴)))
64, 5xrneq12i 37888 . . 3 ((𝑅𝐴) ⋉ (𝑆𝐴)) = ((𝑅 ∩ (𝐴 × ran (𝑅𝐴))) ⋉ (𝑆 ∩ (𝐴 × ran (𝑆𝐴))))
7 simp1 1133 . . . 4 ((𝐴𝑉𝑅𝑊 ∧ (𝑆𝐴) ∈ 𝑋) → 𝐴𝑉)
8 resexg 6036 . . . . . 6 (𝑅𝑊 → (𝑅𝐴) ∈ V)
9 rnexg 7916 . . . . . 6 ((𝑅𝐴) ∈ V → ran (𝑅𝐴) ∈ V)
108, 9syl 17 . . . . 5 (𝑅𝑊 → ran (𝑅𝐴) ∈ V)
11103ad2ant2 1131 . . . 4 ((𝐴𝑉𝑅𝑊 ∧ (𝑆𝐴) ∈ 𝑋) → ran (𝑅𝐴) ∈ V)
12 rnexg 7916 . . . . 5 ((𝑆𝐴) ∈ 𝑋 → ran (𝑆𝐴) ∈ V)
13123ad2ant3 1132 . . . 4 ((𝐴𝑉𝑅𝑊 ∧ (𝑆𝐴) ∈ 𝑋) → ran (𝑆𝐴) ∈ V)
14 inxpxrn 37899 . . . . 5 ((𝑅 ∩ (𝐴 × ran (𝑅𝐴))) ⋉ (𝑆 ∩ (𝐴 × ran (𝑆𝐴)))) = ((𝑅𝑆) ∩ (𝐴 × (ran (𝑅𝐴) × ran (𝑆𝐴))))
15 xrninxpex 37898 . . . . 5 ((𝐴𝑉 ∧ ran (𝑅𝐴) ∈ V ∧ ran (𝑆𝐴) ∈ V) → ((𝑅𝑆) ∩ (𝐴 × (ran (𝑅𝐴) × ran (𝑆𝐴)))) ∈ V)
1614, 15eqeltrid 2833 . . . 4 ((𝐴𝑉 ∧ ran (𝑅𝐴) ∈ V ∧ ran (𝑆𝐴) ∈ V) → ((𝑅 ∩ (𝐴 × ran (𝑅𝐴))) ⋉ (𝑆 ∩ (𝐴 × ran (𝑆𝐴)))) ∈ V)
177, 11, 13, 16syl3anc 1368 . . 3 ((𝐴𝑉𝑅𝑊 ∧ (𝑆𝐴) ∈ 𝑋) → ((𝑅 ∩ (𝐴 × ran (𝑅𝐴))) ⋉ (𝑆 ∩ (𝐴 × ran (𝑆𝐴)))) ∈ V)
186, 17eqeltrid 2833 . 2 ((𝐴𝑉𝑅𝑊 ∧ (𝑆𝐴) ∈ 𝑋) → ((𝑅𝐴) ⋉ (𝑆𝐴)) ∈ V)
193, 18eqeltrrid 2834 1 ((𝐴𝑉𝑅𝑊 ∧ (𝑆𝐴) ∈ 𝑋) → (𝑅 ⋉ (𝑆𝐴)) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084  wcel 2098  Vcvv 3473  cin 3948   × cxp 5680  ran crn 5683  cres 5684  cxrn 37680
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fo 6559  df-fv 6561  df-1st 7999  df-2nd 8000  df-xrn 37875
This theorem is referenced by:  xrnidresex  37911  xrncnvepresex  37912
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