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Theorem uniiunlem 3028
 Description: A subset relationship useful for converting union to indexed union using dfiun2 or dfiun2g and intersection to indexed intersection using dfiin2 . (Contributed by NM, 5-Oct-2006.) (Proof shortened by Mario Carneiro, 26-Sep-2015.)
Assertion
Ref Expression
uniiunlem
Distinct variable groups:   ,   ,   ,   ,
Allowed substitution hints:   ()   ()   ()   (,)

Proof of Theorem uniiunlem
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2046 . . . . . 6
21rexbidv 2327 . . . . 5
32cbvabv 2161 . . . 4
43sseq1i 2969 . . 3
5 r19.23v 2425 . . . . 5
65albii 1359 . . . 4
7 ralcom4 2576 . . . 4
8 abss 3009 . . . 4
96, 7, 83bitr4i 201 . . 3
104, 9bitr4i 176 . 2
11 nfv 1421 . . . . 5
12 eleq1 2100 . . . . 5
1311, 12ceqsalg 2582 . . . 4
1413ralimi 2384 . . 3
15 ralbi 2445 . . 3
1614, 15syl 14 . 2
1710, 16syl5rbb 182 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 98  wal 1241   wceq 1243   wcel 1393  cab 2026  wral 2306  wrex 2307   wss 2917 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-in 2924  df-ss 2931 This theorem is referenced by: (None)
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