Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > mptelixpg | Structured version Visualization version 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 3212 | . 2 | |
2 | nfcv 2764 | . . . . . 6 | |
3 | nfcsb1v 3549 | . . . . . 6 | |
4 | csbeq1a 3542 | . . . . . 6 | |
5 | 2, 3, 4 | cbvixp 7925 | . . . . 5 |
6 | 5 | eleq2i 2693 | . . . 4 |
7 | elixp2 7912 | . . . 4 | |
8 | 3anass 1042 | . . . 4 | |
9 | 6, 7, 8 | 3bitri 286 | . . 3 |
10 | eqid 2622 | . . . . . . . 8 | |
11 | 10 | fnmpt 6020 | . . . . . . 7 |
12 | 10 | fvmpt2 6291 | . . . . . . . . 9 |
13 | simpr 477 | . . . . . . . . 9 | |
14 | 12, 13 | eqeltrd 2701 | . . . . . . . 8 |
15 | 14 | ralimiaa 2951 | . . . . . . 7 |
16 | 11, 15 | jca 554 | . . . . . 6 |
17 | dffn2 6047 | . . . . . . . 8 | |
18 | 10 | fmpt 6381 | . . . . . . . . 9 |
19 | 10 | fvmpt2 6291 | . . . . . . . . . . . . 13 |
20 | 19 | eleq1d 2686 | . . . . . . . . . . . 12 |
21 | 20 | biimpd 219 | . . . . . . . . . . 11 |
22 | 21 | ralimiaa 2951 | . . . . . . . . . 10 |
23 | ralim 2948 | . . . . . . . . . 10 | |
24 | 22, 23 | syl 17 | . . . . . . . . 9 |
25 | 18, 24 | sylbir 225 | . . . . . . . 8 |
26 | 17, 25 | sylbi 207 | . . . . . . 7 |
27 | 26 | imp 445 | . . . . . 6 |
28 | 16, 27 | impbii 199 | . . . . 5 |
29 | nfv 1843 | . . . . . . 7 | |
30 | nffvmpt1 6199 | . . . . . . . 8 | |
31 | 30, 3 | nfel 2777 | . . . . . . 7 |
32 | fveq2 6191 | . . . . . . . 8 | |
33 | 32, 4 | eleq12d 2695 | . . . . . . 7 |
34 | 29, 31, 33 | cbvral 3167 | . . . . . 6 |
35 | 34 | anbi2i 730 | . . . . 5 |
36 | 28, 35 | bitri 264 | . . . 4 |
37 | mptexg 6484 | . . . . 5 | |
38 | 37 | biantrurd 529 | . . . 4 |
39 | 36, 38 | syl5rbb 273 | . . 3 |
40 | 9, 39 | syl5bb 272 | . 2 |
41 | 1, 40 | syl 17 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wcel 1990 wral 2912 cvv 3200 csb 3533 cmpt 4729 wfn 5883 wf 5884 cfv 5888 cixp 7908 |
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-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
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-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 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-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-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ixp 7909 |
This theorem is referenced by: resixpfo 7946 ixpiunwdom 8496 dfac9 8958 prdsbasmpt 16130 prdsbasmpt2 16142 idfucl 16541 fuccocl 16624 fucidcl 16625 invfuc 16634 curf2cl 16871 yonedalem4c 16917 dprdwd 18410 ptpjopn 21415 dfac14lem 21420 ptcnplem 21424 ptcnp 21425 ptcn 21430 ptcmplem2 21857 tmdgsum2 21900 upixp 33524 kelac1 37633 |
Copyright terms: Public domain | W3C validator |