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Theorem riinint 4536
 Description: Express a relative indexed intersection as an intersection. (Contributed by Stefan O'Rear, 22-Feb-2015.)
Assertion
Ref Expression
riinint ((𝑋 𝑉 𝑘 𝐼 𝑆𝑋) → (𝑋 𝑘 𝐼 𝑆) = ({𝑋} ∪ ran (𝑘 𝐼𝑆)))
Distinct variable groups:   𝑘,𝑉   𝑘,𝑋
Allowed substitution hints:   𝑆(𝑘)   𝐼(𝑘)

Proof of Theorem riinint
StepHypRef Expression
1 ssexg 3887 . . . . . . 7 ((𝑆𝑋 𝑋 𝑉) → 𝑆 V)
21expcom 109 . . . . . 6 (𝑋 𝑉 → (𝑆𝑋𝑆 V))
32ralimdv 2382 . . . . 5 (𝑋 𝑉 → (𝑘 𝐼 𝑆𝑋𝑘 𝐼 𝑆 V))
43imp 115 . . . 4 ((𝑋 𝑉 𝑘 𝐼 𝑆𝑋) → 𝑘 𝐼 𝑆 V)
5 dfiin3g 4533 . . . 4 (𝑘 𝐼 𝑆 V → 𝑘 𝐼 𝑆 = ran (𝑘 𝐼𝑆))
64, 5syl 14 . . 3 ((𝑋 𝑉 𝑘 𝐼 𝑆𝑋) → 𝑘 𝐼 𝑆 = ran (𝑘 𝐼𝑆))
76ineq2d 3132 . 2 ((𝑋 𝑉 𝑘 𝐼 𝑆𝑋) → (𝑋 𝑘 𝐼 𝑆) = (𝑋 ran (𝑘 𝐼𝑆)))
8 intun 3637 . . 3 ({𝑋} ∪ ran (𝑘 𝐼𝑆)) = ( {𝑋} ∩ ran (𝑘 𝐼𝑆))
9 intsng 3640 . . . . 5 (𝑋 𝑉 {𝑋} = 𝑋)
109adantr 261 . . . 4 ((𝑋 𝑉 𝑘 𝐼 𝑆𝑋) → {𝑋} = 𝑋)
1110ineq1d 3131 . . 3 ((𝑋 𝑉 𝑘 𝐼 𝑆𝑋) → ( {𝑋} ∩ ran (𝑘 𝐼𝑆)) = (𝑋 ran (𝑘 𝐼𝑆)))
128, 11syl5eq 2081 . 2 ((𝑋 𝑉 𝑘 𝐼 𝑆𝑋) → ({𝑋} ∪ ran (𝑘 𝐼𝑆)) = (𝑋 ran (𝑘 𝐼𝑆)))
137, 12eqtr4d 2072 1 ((𝑋 𝑉 𝑘 𝐼 𝑆𝑋) → (𝑋 𝑘 𝐼 𝑆) = ({𝑋} ∪ ran (𝑘 𝐼𝑆)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242   ∈ wcel 1390  ∀wral 2300  Vcvv 2551   ∪ cun 2909   ∩ cin 2910   ⊆ wss 2911  {csn 3367  ∩ cint 3606  ∩ ciin 3649   ↦ cmpt 3809  ran crn 4289 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-int 3607  df-iin 3651  df-br 3756  df-opab 3810  df-mpt 3811  df-cnv 4296  df-dm 4298  df-rn 4299 This theorem is referenced by: (None)
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