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Mirrors > Home > MPE Home > Th. List > coiun | Unicode version |
Description: Composition with an indexed union. (Contributed by NM, 21-Dec-2008.) |
Ref | Expression |
---|---|
coiun |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relco 5510 | . 2 | |
2 | reliun 5128 | . . 3 | |
3 | relco 5510 | . . . 4 | |
4 | 3 | a1i 11 | . . 3 |
5 | 2, 4 | mprgbir 2821 | . 2 |
6 | eliun 4335 | . . . . . . . 8 | |
7 | df-br 4453 | . . . . . . . 8 | |
8 | df-br 4453 | . . . . . . . . 9 | |
9 | 8 | rexbii 2959 | . . . . . . . 8 |
10 | 6, 7, 9 | 3bitr4i 277 | . . . . . . 7 |
11 | 10 | anbi1i 695 | . . . . . 6 |
12 | r19.41v 3009 | . . . . . 6 | |
13 | 11, 12 | bitr4i 252 | . . . . 5 |
14 | 13 | exbii 1667 | . . . 4 |
15 | rexcom4 3129 | . . . 4 | |
16 | 14, 15 | bitr4i 252 | . . 3 |
17 | vex 3112 | . . . 4 | |
18 | vex 3112 | . . . 4 | |
19 | 17, 18 | opelco 5179 | . . 3 |
20 | eliun 4335 | . . . 4 | |
21 | 17, 18 | opelco 5179 | . . . . 5 |
22 | 21 | rexbii 2959 | . . . 4 |
23 | 20, 22 | bitri 249 | . . 3 |
24 | 16, 19, 23 | 3bitr4i 277 | . 2 |
25 | 1, 5, 24 | eqrelriiv 5102 | 1 |
Colors of variables: wff setvar class |
Syntax hints: /\ wa 369 = wceq 1395
E. wex 1612 e. wcel 1818 E. wrex 2808
<. cop 4035 U_ ciun 4330 class class class wbr 4452
o. ccom 5008 Rel wrel 5009 |
This theorem is referenced by: fparlem3 6902 fparlem4 6903 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1618 ax-4 1631 ax-5 1704 ax-6 1747 ax-7 1790 ax-9 1822 ax-10 1837 ax-11 1842 ax-12 1854 ax-13 1999 ax-ext 2435 ax-sep 4573 ax-nul 4581 ax-pr 4691 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 975 df-tru 1398 df-ex 1613 df-nf 1617 df-sb 1740 df-eu 2286 df-mo 2287 df-clab 2443 df-cleq 2449 df-clel 2452 df-nfc 2607 df-ne 2654 df-ral 2812 df-rex 2813 df-rab 2816 df-v 3111 df-dif 3478 df-un 3480 df-in 3482 df-ss 3489 df-nul 3785 df-if 3942 df-sn 4030 df-pr 4032 df-op 4036 df-iun 4332 df-br 4453 df-opab 4511 df-xp 5010 df-rel 5011 df-co 5013 |
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