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Theorem dfiun2g 4955
Description: Alternate definition of indexed union when 𝐵 is a set. Definition 15(a) of [Suppes] p. 44. (Contributed by NM, 23-Mar-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (Proof shortened by Rohan Ridenour, 11-Aug-2023.)
Assertion
Ref Expression
dfiun2g (∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥,𝑦)

Proof of Theorem dfiun2g
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfra1 3141 . . . . . 6 𝑥𝑥𝐴 𝐵𝐶
2 rspa 3129 . . . . . . 7 ((∀𝑥𝐴 𝐵𝐶𝑥𝐴) → 𝐵𝐶)
3 clel3g 3582 . . . . . . 7 (𝐵𝐶 → (𝑧𝐵 ↔ ∃𝑦(𝑦 = 𝐵𝑧𝑦)))
42, 3syl 17 . . . . . 6 ((∀𝑥𝐴 𝐵𝐶𝑥𝐴) → (𝑧𝐵 ↔ ∃𝑦(𝑦 = 𝐵𝑧𝑦)))
51, 4rexbida 3245 . . . . 5 (∀𝑥𝐴 𝐵𝐶 → (∃𝑥𝐴 𝑧𝐵 ↔ ∃𝑥𝐴𝑦(𝑦 = 𝐵𝑧𝑦)))
6 rexcom4 3178 . . . . 5 (∃𝑥𝐴𝑦(𝑦 = 𝐵𝑧𝑦) ↔ ∃𝑦𝑥𝐴 (𝑦 = 𝐵𝑧𝑦))
75, 6bitrdi 290 . . . 4 (∀𝑥𝐴 𝐵𝐶 → (∃𝑥𝐴 𝑧𝐵 ↔ ∃𝑦𝑥𝐴 (𝑦 = 𝐵𝑧𝑦)))
8 r19.41v 3273 . . . . . 6 (∃𝑥𝐴 (𝑦 = 𝐵𝑧𝑦) ↔ (∃𝑥𝐴 𝑦 = 𝐵𝑧𝑦))
98exbii 1855 . . . . 5 (∃𝑦𝑥𝐴 (𝑦 = 𝐵𝑧𝑦) ↔ ∃𝑦(∃𝑥𝐴 𝑦 = 𝐵𝑧𝑦))
10 exancom 1869 . . . . 5 (∃𝑦(∃𝑥𝐴 𝑦 = 𝐵𝑧𝑦) ↔ ∃𝑦(𝑧𝑦 ∧ ∃𝑥𝐴 𝑦 = 𝐵))
119, 10bitri 278 . . . 4 (∃𝑦𝑥𝐴 (𝑦 = 𝐵𝑧𝑦) ↔ ∃𝑦(𝑧𝑦 ∧ ∃𝑥𝐴 𝑦 = 𝐵))
127, 11bitrdi 290 . . 3 (∀𝑥𝐴 𝐵𝐶 → (∃𝑥𝐴 𝑧𝐵 ↔ ∃𝑦(𝑧𝑦 ∧ ∃𝑥𝐴 𝑦 = 𝐵)))
13 eliun 4923 . . 3 (𝑧 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑧𝐵)
14 eluniab 4849 . . 3 (𝑧 {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ↔ ∃𝑦(𝑧𝑦 ∧ ∃𝑥𝐴 𝑦 = 𝐵))
1512, 13, 143bitr4g 317 . 2 (∀𝑥𝐴 𝐵𝐶 → (𝑧 𝑥𝐴 𝐵𝑧 {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}))
1615eqrdv 2736 1 (∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wex 1787  wcel 2111  {cab 2715  wral 3062  wrex 3063   cuni 4834   ciun 4919
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2072  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3067  df-rex 3068  df-v 3423  df-uni 4835  df-iun 4921
This theorem is referenced by:  dfiun2  4957  dfiun3g  5848  abnexg  7560  iunexg  7755  uniqs  8480  ac6num  10118  iunopn  21822  pnrmopn  22267  cncmp  22316  ptcmplem3  22978  iunmbl  24477  voliun  24478  sigaclcuni  31825  sigaclcu2  31827  sigaclci  31839  measvunilem  31919  meascnbl  31926  carsgclctunlem3  32026  uniqsALTV  36231
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