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Theorem iunxsng 3723
Description: A singleton index picks out an instance of an indexed union's argument. (Contributed by Mario Carneiro, 25-Jun-2016.)
Hypothesis
Ref Expression
iunxsng.1 (x = AB = 𝐶)
Assertion
Ref Expression
iunxsng (A 𝑉 x {A}B = 𝐶)
Distinct variable groups:   x,A   x,𝐶
Allowed substitution hints:   B(x)   𝑉(x)

Proof of Theorem iunxsng
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 eliun 3652 . . 3 (y x {A}Bx {A}y B)
2 iunxsng.1 . . . . 5 (x = AB = 𝐶)
32eleq2d 2104 . . . 4 (x = A → (y By 𝐶))
43rexsng 3403 . . 3 (A 𝑉 → (x {A}y By 𝐶))
51, 4syl5bb 181 . 2 (A 𝑉 → (y x {A}By 𝐶))
65eqrdv 2035 1 (A 𝑉 x {A}B = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242   wcel 1390  wrex 2301  {csn 3367   ciun 3648
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-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-sn 3373  df-iun 3650
This theorem is referenced by:  iunxsn  3724  rdgisuc1  5911  oasuc  5983  omsuc  5990
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