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Theorem dfiun2g 3745
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.)
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 2405 . . . . . 6 𝑥𝑥𝐴 𝐵𝐶
2 rsp 2419 . . . . . . . 8 (∀𝑥𝐴 𝐵𝐶 → (𝑥𝐴𝐵𝐶))
3 clel3g 2742 . . . . . . . 8 (𝐵𝐶 → (𝑧𝐵 ↔ ∃𝑦(𝑦 = 𝐵𝑧𝑦)))
42, 3syl6 33 . . . . . . 7 (∀𝑥𝐴 𝐵𝐶 → (𝑥𝐴 → (𝑧𝐵 ↔ ∃𝑦(𝑦 = 𝐵𝑧𝑦))))
54imp 122 . . . . . 6 ((∀𝑥𝐴 𝐵𝐶𝑥𝐴) → (𝑧𝐵 ↔ ∃𝑦(𝑦 = 𝐵𝑧𝑦)))
61, 5rexbida 2371 . . . . 5 (∀𝑥𝐴 𝐵𝐶 → (∃𝑥𝐴 𝑧𝐵 ↔ ∃𝑥𝐴𝑦(𝑦 = 𝐵𝑧𝑦)))
7 rexcom4 2636 . . . . 5 (∃𝑥𝐴𝑦(𝑦 = 𝐵𝑧𝑦) ↔ ∃𝑦𝑥𝐴 (𝑦 = 𝐵𝑧𝑦))
86, 7syl6bb 194 . . . 4 (∀𝑥𝐴 𝐵𝐶 → (∃𝑥𝐴 𝑧𝐵 ↔ ∃𝑦𝑥𝐴 (𝑦 = 𝐵𝑧𝑦)))
9 r19.41v 2519 . . . . . 6 (∃𝑥𝐴 (𝑦 = 𝐵𝑧𝑦) ↔ (∃𝑥𝐴 𝑦 = 𝐵𝑧𝑦))
109exbii 1539 . . . . 5 (∃𝑦𝑥𝐴 (𝑦 = 𝐵𝑧𝑦) ↔ ∃𝑦(∃𝑥𝐴 𝑦 = 𝐵𝑧𝑦))
11 exancom 1542 . . . . 5 (∃𝑦(∃𝑥𝐴 𝑦 = 𝐵𝑧𝑦) ↔ ∃𝑦(𝑧𝑦 ∧ ∃𝑥𝐴 𝑦 = 𝐵))
1210, 11bitri 182 . . . 4 (∃𝑦𝑥𝐴 (𝑦 = 𝐵𝑧𝑦) ↔ ∃𝑦(𝑧𝑦 ∧ ∃𝑥𝐴 𝑦 = 𝐵))
138, 12syl6bb 194 . . 3 (∀𝑥𝐴 𝐵𝐶 → (∃𝑥𝐴 𝑧𝐵 ↔ ∃𝑦(𝑧𝑦 ∧ ∃𝑥𝐴 𝑦 = 𝐵)))
14 eliun 3717 . . 3 (𝑧 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑧𝐵)
15 eluniab 3648 . . 3 (𝑧 {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ↔ ∃𝑦(𝑧𝑦 ∧ ∃𝑥𝐴 𝑦 = 𝐵))
1613, 14, 153bitr4g 221 . 2 (∀𝑥𝐴 𝐵𝐶 → (𝑧 𝑥𝐴 𝐵𝑧 {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}))
1716eqrdv 2083 1 (∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1287  wex 1424  wcel 1436  {cab 2071  wral 2355  wrex 2356   cuni 3636   ciun 3713
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-uni 3637  df-iun 3715
This theorem is referenced by:  dfiun2  3747  dfiun3g  4657  fniunfv  5496  iunexg  5841  uniqs  6296
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