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Theorem reliun 4401
Description: An indexed union is a relation iff each member of its indexed family is a relation. (Contributed by NM, 19-Dec-2008.)
Assertion
Ref Expression
reliun (Rel x A Bx A Rel B)

Proof of Theorem reliun
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 df-iun 3650 . . 3 x A B = {yx A y B}
21releqi 4366 . 2 (Rel x A B ↔ Rel {yx A y B})
3 df-rel 4295 . 2 (Rel {yx A y B} ↔ {yx A y B} ⊆ (V × V))
4 abss 3003 . . 3 ({yx A y B} ⊆ (V × V) ↔ y(x A y By (V × V)))
5 df-rel 4295 . . . . . 6 (Rel BB ⊆ (V × V))
6 dfss2 2928 . . . . . 6 (B ⊆ (V × V) ↔ y(y By (V × V)))
75, 6bitri 173 . . . . 5 (Rel By(y By (V × V)))
87ralbii 2324 . . . 4 (x A Rel Bx A y(y By (V × V)))
9 ralcom4 2570 . . . 4 (x A y(y By (V × V)) ↔ yx A (y By (V × V)))
10 r19.23v 2419 . . . . 5 (x A (y By (V × V)) ↔ (x A y By (V × V)))
1110albii 1356 . . . 4 (yx A (y By (V × V)) ↔ y(x A y By (V × V)))
128, 9, 113bitri 195 . . 3 (x A Rel By(x A y By (V × V)))
134, 12bitr4i 176 . 2 ({yx A y B} ⊆ (V × V) ↔ x A Rel B)
142, 3, 133bitri 195 1 (Rel x A Bx A Rel B)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1240   wcel 1390  {cab 2023  wral 2300  wrex 2301  Vcvv 2551  wss 2911   ciun 3648   × cxp 4286  Rel wrel 4293
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-in 2918  df-ss 2925  df-iun 3650  df-rel 4295
This theorem is referenced by:  reluni  4403  eliunxp  4418  opeliunxp2  4419  dfco2  4763  coiun  4773
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