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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 2732 | . 2 | |
2 | nfcv 2306 | . . . . . 6 | |
3 | nfcsb1v 3073 | . . . . . 6 | |
4 | csbeq1a 3049 | . . . . . 6 | |
5 | 2, 3, 4 | cbvixp 6672 | . . . . 5 |
6 | 5 | eleq2i 2231 | . . . 4 |
7 | elixp2 6659 | . . . 4 | |
8 | 3anass 971 | . . . 4 | |
9 | 6, 7, 8 | 3bitri 205 | . . 3 |
10 | eqid 2164 | . . . . . . . 8 | |
11 | 10 | fnmpt 5308 | . . . . . . 7 |
12 | 10 | fvmpt2 5563 | . . . . . . . . 9 |
13 | simpr 109 | . . . . . . . . 9 | |
14 | 12, 13 | eqeltrd 2241 | . . . . . . . 8 |
15 | 14 | ralimiaa 2526 | . . . . . . 7 |
16 | 11, 15 | jca 304 | . . . . . 6 |
17 | dffn2 5333 | . . . . . . . 8 | |
18 | 10 | fmpt 5629 | . . . . . . . . 9 |
19 | 10 | fvmpt2 5563 | . . . . . . . . . . . . 13 |
20 | 19 | eleq1d 2233 | . . . . . . . . . . . 12 |
21 | 20 | biimpd 143 | . . . . . . . . . . 11 |
22 | 21 | ralimiaa 2526 | . . . . . . . . . 10 |
23 | ralim 2523 | . . . . . . . . . 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 1515 | . . . . . . 7 | |
30 | nffvmpt1 5491 | . . . . . . . 8 | |
31 | 30, 3 | nfel 2315 | . . . . . . 7 |
32 | fveq2 5480 | . . . . . . . 8 | |
33 | 32, 4 | eleq12d 2235 | . . . . . . 7 |
34 | 29, 31, 33 | cbvral 2685 | . . . . . 6 |
35 | 34 | anbi2i 453 | . . . . 5 |
36 | 28, 35 | bitri 183 | . . . 4 |
37 | mptexg 5704 | . . . . 5 | |
38 | 37 | biantrurd 303 | . . . 4 |
39 | 36, 38 | bitr2id 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 967 wcel 2135 wral 2442 cvv 2721 csb 3040 cmpt 4037 wfn 5177 wf 5178 cfv 5182 cixp 6655 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ixp 6656 |
This theorem is referenced by: (None) |
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