Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  xrnresex Structured version   Visualization version   GIF version

Theorem xrnresex 34650
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 34648 . . 3 ((𝑅𝑆) ↾ 𝐴) = ((𝑅𝐴) ⋉ (𝑆𝐴))
2 xrnres2 34647 . . 3 ((𝑅𝑆) ↾ 𝐴) = (𝑅 ⋉ (𝑆𝐴))
31, 2eqtr3i 2821 . 2 ((𝑅𝐴) ⋉ (𝑆𝐴)) = (𝑅 ⋉ (𝑆𝐴))
4 dfres4 34550 . . . 4 (𝑅𝐴) = (𝑅 ∩ (𝐴 × ran (𝑅𝐴)))
5 dfres4 34550 . . . 4 (𝑆𝐴) = (𝑆 ∩ (𝐴 × ran (𝑆𝐴)))
64, 5xrneq12i 34632 . . 3 ((𝑅𝐴) ⋉ (𝑆𝐴)) = ((𝑅 ∩ (𝐴 × ran (𝑅𝐴))) ⋉ (𝑆 ∩ (𝐴 × ran (𝑆𝐴))))
7 simp1 1167 . . . 4 ((𝐴𝑉𝑅𝑊 ∧ (𝑆𝐴) ∈ 𝑋) → 𝐴𝑉)
8 resexg 5652 . . . . . 6 (𝑅𝑊 → (𝑅𝐴) ∈ V)
9 rnexg 7330 . . . . . 6 ((𝑅𝐴) ∈ V → ran (𝑅𝐴) ∈ V)
108, 9syl 17 . . . . 5 (𝑅𝑊 → ran (𝑅𝐴) ∈ V)
11103ad2ant2 1165 . . . 4 ((𝐴𝑉𝑅𝑊 ∧ (𝑆𝐴) ∈ 𝑋) → ran (𝑅𝐴) ∈ V)
12 rnexg 7330 . . . . 5 ((𝑆𝐴) ∈ 𝑋 → ran (𝑆𝐴) ∈ V)
13123ad2ant3 1166 . . . 4 ((𝐴𝑉𝑅𝑊 ∧ (𝑆𝐴) ∈ 𝑋) → ran (𝑆𝐴) ∈ V)
14 inxpxrn 34639 . . . . 5 ((𝑅 ∩ (𝐴 × ran (𝑅𝐴))) ⋉ (𝑆 ∩ (𝐴 × ran (𝑆𝐴)))) = ((𝑅𝑆) ∩ (𝐴 × (ran (𝑅𝐴) × ran (𝑆𝐴))))
15 xrninxpex 34638 . . . . 5 ((𝐴𝑉 ∧ ran (𝑅𝐴) ∈ V ∧ ran (𝑆𝐴) ∈ V) → ((𝑅𝑆) ∩ (𝐴 × (ran (𝑅𝐴) × ran (𝑆𝐴)))) ∈ V)
1614, 15syl5eqel 2880 . . . 4 ((𝐴𝑉 ∧ ran (𝑅𝐴) ∈ V ∧ ran (𝑆𝐴) ∈ V) → ((𝑅 ∩ (𝐴 × ran (𝑅𝐴))) ⋉ (𝑆 ∩ (𝐴 × ran (𝑆𝐴)))) ∈ V)
177, 11, 13, 16syl3anc 1491 . . 3 ((𝐴𝑉𝑅𝑊 ∧ (𝑆𝐴) ∈ 𝑋) → ((𝑅 ∩ (𝐴 × ran (𝑅𝐴))) ⋉ (𝑆 ∩ (𝐴 × ran (𝑆𝐴)))) ∈ V)
186, 17syl5eqel 2880 . 2 ((𝐴𝑉𝑅𝑊 ∧ (𝑆𝐴) ∈ 𝑋) → ((𝑅𝐴) ⋉ (𝑆𝐴)) ∈ V)
193, 18syl5eqelr 2881 1 ((𝐴𝑉𝑅𝑊 ∧ (𝑆𝐴) ∈ 𝑋) → (𝑅 ⋉ (𝑆𝐴)) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1108  wcel 2157  Vcvv 3383  cin 3766   × cxp 5308  ran crn 5311  cres 5312  cxrn 34460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-fo 6105  df-fv 6107  df-1st 7399  df-2nd 7400  df-xrn 34619
This theorem is referenced by:  xrnidresex  34651  xrncnvepresex  34652
  Copyright terms: Public domain W3C validator