Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > resixp | Unicode version |
Description: Restriction of an element of an infinite Cartesian product. (Contributed by FL, 7-Nov-2011.) (Proof shortened by Mario Carneiro, 31-May-2014.) |
Ref | Expression |
---|---|
resixp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resexg 4918 | . . 3 | |
2 | 1 | adantl 275 | . 2 |
3 | simpr 109 | . . . . 5 | |
4 | elixp2 6659 | . . . . 5 | |
5 | 3, 4 | sylib 121 | . . . 4 |
6 | 5 | simp2d 999 | . . 3 |
7 | simpl 108 | . . 3 | |
8 | fnssres 5295 | . . 3 | |
9 | 6, 7, 8 | syl2anc 409 | . 2 |
10 | 5 | simp3d 1000 | . . . 4 |
11 | ssralv 3201 | . . . 4 | |
12 | 7, 10, 11 | sylc 62 | . . 3 |
13 | fvres 5504 | . . . . 5 | |
14 | 13 | eleq1d 2233 | . . . 4 |
15 | 14 | ralbiia 2478 | . . 3 |
16 | 12, 15 | sylibr 133 | . 2 |
17 | elixp2 6659 | . 2 | |
18 | 2, 9, 16, 17 | syl3anbrc 1170 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 967 wcel 2135 wral 2442 cvv 2721 wss 3111 cres 4600 wfn 5177 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-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-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 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-br 3977 df-opab 4038 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-res 4610 df-iota 5147 df-fun 5184 df-fn 5185 df-fv 5190 df-ixp 6656 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |