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Theorem dfiun2g 3680
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 (x A B 𝐶 x A B = {yx A 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 2349 . . . . . 6 xx A B 𝐶
2 rsp 2363 . . . . . . . 8 (x A B 𝐶 → (x AB 𝐶))
3 clel3g 2672 . . . . . . . 8 (B 𝐶 → (z By(y = B z y)))
42, 3syl6 29 . . . . . . 7 (x A B 𝐶 → (x A → (z By(y = B z y))))
54imp 115 . . . . . 6 ((x A B 𝐶 x A) → (z By(y = B z y)))
61, 5rexbida 2315 . . . . 5 (x A B 𝐶 → (x A z Bx A y(y = B z y)))
7 rexcom4 2571 . . . . 5 (x A y(y = B z y) ↔ yx A (y = B z y))
86, 7syl6bb 185 . . . 4 (x A B 𝐶 → (x A z Byx A (y = B z y)))
9 r19.41v 2460 . . . . . 6 (x A (y = B z y) ↔ (x A y = B z y))
109exbii 1493 . . . . 5 (yx A (y = B z y) ↔ y(x A y = B z y))
11 exancom 1496 . . . . 5 (y(x A y = B z y) ↔ y(z y x A y = B))
1210, 11bitri 173 . . . 4 (yx A (y = B z y) ↔ y(z y x A y = B))
138, 12syl6bb 185 . . 3 (x A B 𝐶 → (x A z By(z y x A y = B)))
14 eliun 3652 . . 3 (z x A Bx A z B)
15 eluniab 3583 . . 3 (z {yx A y = B} ↔ y(z y x A y = B))
1613, 14, 153bitr4g 212 . 2 (x A B 𝐶 → (z x A Bz {yx A y = B}))
1716eqrdv 2035 1 (x A B 𝐶 x A B = {yx A y = B})
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  wex 1378   wcel 1390  {cab 2023  wral 2300  wrex 2301   cuni 3571   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-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-uni 3572  df-iun 3650
This theorem is referenced by:  dfiun2  3682  dfiun3g  4532  fniunfv  5344  iunexg  5688  uniqs  6100
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