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Theorem gruiun 8666
 Description: If is a family of elements of and the index set is an element of , then the indexed union is also an element of , where is a Grothendieck's universe. (Contributed by Mario Carneiro, 9-Jun-2013.)
Assertion
Ref Expression
gruiun
Distinct variable groups:   ,   ,
Allowed substitution hint:   ()

Proof of Theorem gruiun
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eqid 2435 . . . . . . 7
21fnmpt 5563 . . . . . 6
31rnmpt 5108 . . . . . . 7
4 r19.29 2838 . . . . . . . . . 10
5 eleq1 2495 . . . . . . . . . . . 12
65biimparc 474 . . . . . . . . . . 11
76rexlimivw 2818 . . . . . . . . . 10
84, 7syl 16 . . . . . . . . 9
98ex 424 . . . . . . . 8
109abssdv 3409 . . . . . . 7
113, 10syl5eqss 3384 . . . . . 6
12 df-f 5450 . . . . . 6
132, 11, 12sylanbrc 646 . . . . 5
14 gruurn 8665 . . . . . 6
15143expia 1155 . . . . 5
1613, 15syl5com 28 . . . 4
17 dfiun3g 5114 . . . . 5
1817eleq1d 2501 . . . 4
1916, 18sylibrd 226 . . 3
2019com12 29 . 2
21203impia 1150 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   w3a 936   wceq 1652   wcel 1725  cab 2421  wral 2697  wrex 2698   wss 3312  cuni 4007  ciun 4085   cmpt 4258   crn 4871   wfn 5441  wf 5442  cgru 8657 This theorem is referenced by:  gruuni  8667  gruun  8673  gruixp  8676  grur1a  8686 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-map 7012  df-gru 8658
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