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Mirrors > Home > MPE Home > Th. List > gruiun | Structured version Visualization version Unicode version |
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 universe. (Contributed by Mario Carneiro, 9-Jun-2013.) |
Ref | Expression |
---|---|
gruiun |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . . . . . 7 | |
2 | 1 | fnmpt 6020 | . . . . . 6 |
3 | 1 | rnmptss 6392 | . . . . . 6 |
4 | df-f 5892 | . . . . . 6 | |
5 | 2, 3, 4 | sylanbrc 698 | . . . . 5 |
6 | gruurn 9620 | . . . . . 6 | |
7 | 6 | 3expia 1267 | . . . . 5 |
8 | 5, 7 | syl5com 31 | . . . 4 |
9 | dfiun3g 5378 | . . . . 5 | |
10 | 9 | eleq1d 2686 | . . . 4 |
11 | 8, 10 | sylibrd 249 | . . 3 |
12 | 11 | com12 32 | . 2 |
13 | 12 | 3impia 1261 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wcel 1990 wral 2912 wss 3574 cuni 4436 ciun 4520 cmpt 4729 crn 5115 wfn 5883 wf 5884 cgru 9612 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-map 7859 df-gru 9613 |
This theorem is referenced by: gruuni 9622 gruun 9628 gruixp 9631 grur1a 9641 |
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