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Theorem dfiun2g 3663
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 2333 . . . . . 6 xx A B 𝐶
2 rsp 2347 . . . . . . . 8 (x A B 𝐶 → (x AB 𝐶))
3 clel3g 2655 . . . . . . . 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 2299 . . . . 5 (x A B 𝐶 → (x A z Bx A y(y = B z y)))
7 rexcom4 2554 . . . . 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 2444 . . . . . 6 (x A (y = B z y) ↔ (x A y = B z y))
109exbii 1478 . . . . 5 (yx A (y = B z y) ↔ y(x A y = B z y))
11 exancom 1481 . . . . 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 3635 . . 3 (z x A Bx A z B)
15 eluniab 3566 . . 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 2020 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 1228  wex 1362   wcel 1374  {cab 2008  wral 2284  wrex 2285   cuni 3554   ciun 3631
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-uni 3555  df-iun 3633
This theorem is referenced by:  dfiun2  3665  dfiun3g  4516  fniunfv  5326  iunexg  5669  uniqs  6075
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