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Mirrors > Home > MPE Home > Th. List > dfiin2g | Structured version Visualization version Unicode version |
Description: Alternate definition of indexed intersection when is a set. (Contributed by Jeff Hankins, 27-Aug-2009.) |
Ref | Expression |
---|---|
dfiin2g |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ral 2917 | . . . 4 | |
2 | df-ral 2917 | . . . . . 6 | |
3 | eleq2 2690 | . . . . . . . . . . . . 13 | |
4 | 3 | biimprcd 240 | . . . . . . . . . . . 12 |
5 | 4 | alrimiv 1855 | . . . . . . . . . . 11 |
6 | eqid 2622 | . . . . . . . . . . . 12 | |
7 | eqeq1 2626 | . . . . . . . . . . . . . 14 | |
8 | 7, 3 | imbi12d 334 | . . . . . . . . . . . . 13 |
9 | 8 | spcgv 3293 | . . . . . . . . . . . 12 |
10 | 6, 9 | mpii 46 | . . . . . . . . . . 11 |
11 | 5, 10 | impbid2 216 | . . . . . . . . . 10 |
12 | 11 | imim2i 16 | . . . . . . . . 9 |
13 | 12 | pm5.74d 262 | . . . . . . . 8 |
14 | 13 | alimi 1739 | . . . . . . 7 |
15 | albi 1746 | . . . . . . 7 | |
16 | 14, 15 | syl 17 | . . . . . 6 |
17 | 2, 16 | sylbi 207 | . . . . 5 |
18 | df-ral 2917 | . . . . . . . 8 | |
19 | 18 | albii 1747 | . . . . . . 7 |
20 | alcom 2037 | . . . . . . 7 | |
21 | 19, 20 | bitr4i 267 | . . . . . 6 |
22 | r19.23v 3023 | . . . . . . . 8 | |
23 | vex 3203 | . . . . . . . . . 10 | |
24 | eqeq1 2626 | . . . . . . . . . . 11 | |
25 | 24 | rexbidv 3052 | . . . . . . . . . 10 |
26 | 23, 25 | elab 3350 | . . . . . . . . 9 |
27 | 26 | imbi1i 339 | . . . . . . . 8 |
28 | 22, 27 | bitr4i 267 | . . . . . . 7 |
29 | 28 | albii 1747 | . . . . . 6 |
30 | 19.21v 1868 | . . . . . . 7 | |
31 | 30 | albii 1747 | . . . . . 6 |
32 | 21, 29, 31 | 3bitr3ri 291 | . . . . 5 |
33 | 17, 32 | syl6bb 276 | . . . 4 |
34 | 1, 33 | syl5bb 272 | . . 3 |
35 | 34 | abbidv 2741 | . 2 |
36 | df-iin 4523 | . 2 | |
37 | df-int 4476 | . 2 | |
38 | 35, 36, 37 | 3eqtr4g 2681 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wal 1481 wceq 1483 wcel 1990 cab 2608 wral 2912 wrex 2913 cint 4475 ciin 4521 |
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-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-v 3202 df-int 4476 df-iin 4523 |
This theorem is referenced by: dfiin2 4555 iinexg 4824 dfiin3g 5379 iinfi 8323 mreiincl 16256 iinopn 20707 clsval2 20854 alexsublem 21848 |
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