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Mirrors > Home > ILE Home > Th. List > mptelixpg | Unicode version |
Description: Condition for an explicit member of an indexed product. (Contributed by Stefan O'Rear, 4-Jan-2015.) |
Ref | Expression |
---|---|
mptelixpg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2697 | . 2 | |
2 | nfcv 2281 | . . . . . 6 | |
3 | nfcsb1v 3035 | . . . . . 6 | |
4 | csbeq1a 3012 | . . . . . 6 | |
5 | 2, 3, 4 | cbvixp 6609 | . . . . 5 |
6 | 5 | eleq2i 2206 | . . . 4 |
7 | elixp2 6596 | . . . 4 | |
8 | 3anass 966 | . . . 4 | |
9 | 6, 7, 8 | 3bitri 205 | . . 3 |
10 | eqid 2139 | . . . . . . . 8 | |
11 | 10 | fnmpt 5249 | . . . . . . 7 |
12 | 10 | fvmpt2 5504 | . . . . . . . . 9 |
13 | simpr 109 | . . . . . . . . 9 | |
14 | 12, 13 | eqeltrd 2216 | . . . . . . . 8 |
15 | 14 | ralimiaa 2494 | . . . . . . 7 |
16 | 11, 15 | jca 304 | . . . . . 6 |
17 | dffn2 5274 | . . . . . . . 8 | |
18 | 10 | fmpt 5570 | . . . . . . . . 9 |
19 | 10 | fvmpt2 5504 | . . . . . . . . . . . . 13 |
20 | 19 | eleq1d 2208 | . . . . . . . . . . . 12 |
21 | 20 | biimpd 143 | . . . . . . . . . . 11 |
22 | 21 | ralimiaa 2494 | . . . . . . . . . 10 |
23 | ralim 2491 | . . . . . . . . . 10 | |
24 | 22, 23 | syl 14 | . . . . . . . . 9 |
25 | 18, 24 | sylbir 134 | . . . . . . . 8 |
26 | 17, 25 | sylbi 120 | . . . . . . 7 |
27 | 26 | imp 123 | . . . . . 6 |
28 | 16, 27 | impbii 125 | . . . . 5 |
29 | nfv 1508 | . . . . . . 7 | |
30 | nffvmpt1 5432 | . . . . . . . 8 | |
31 | 30, 3 | nfel 2290 | . . . . . . 7 |
32 | fveq2 5421 | . . . . . . . 8 | |
33 | 32, 4 | eleq12d 2210 | . . . . . . 7 |
34 | 29, 31, 33 | cbvral 2650 | . . . . . 6 |
35 | 34 | anbi2i 452 | . . . . 5 |
36 | 28, 35 | bitri 183 | . . . 4 |
37 | mptexg 5645 | . . . . 5 | |
38 | 37 | biantrurd 303 | . . . 4 |
39 | 36, 38 | syl5rbb 192 | . . 3 |
40 | 9, 39 | syl5bb 191 | . 2 |
41 | 1, 40 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wcel 1480 wral 2416 cvv 2686 csb 3003 cmpt 3989 wfn 5118 wf 5119 cfv 5123 cixp 6592 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ixp 6593 |
This theorem is referenced by: (None) |
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