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Mirrors > Home > NFE Home > Th. List > imakexg | Unicode version |
Description: The image of a set under a set is a set. (Contributed by SF, 14-Jan-2015.) |
Ref | Expression |
---|---|
imakexg | k |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfimak2 4298 | . 2 k ∼ P6 ∼ 1c k SIk ∼ k | |
2 | 1cex 4142 | . . . . . 6 1c | |
3 | vvex 4109 | . . . . . 6 | |
4 | 2, 3 | xpkex 4289 | . . . . 5 1c k |
5 | 4 | complex 4104 | . . . 4 ∼ 1c k |
6 | xpkexg 4288 | . . . . . . 7 k | |
7 | 3, 6 | mpan2 652 | . . . . . 6 k |
8 | inexg 4100 | . . . . . 6 k k | |
9 | 7, 8 | sylan2 460 | . . . . 5 k |
10 | complexg 4099 | . . . . 5 k ∼ k | |
11 | sikexg 4296 | . . . . 5 ∼ k SIk ∼ k | |
12 | 9, 10, 11 | 3syl 18 | . . . 4 SIk ∼ k |
13 | unexg 4101 | . . . 4 ∼ 1c k SIk ∼ k ∼ 1c k SIk ∼ k | |
14 | 5, 12, 13 | sylancr 644 | . . 3 ∼ 1c k SIk ∼ k |
15 | p6exg 4290 | . . 3 ∼ 1c k SIk ∼ k P6 ∼ 1c k SIk ∼ k | |
16 | complexg 4099 | . . 3 P6 ∼ 1c k SIk ∼ k ∼ P6 ∼ 1c k SIk ∼ k | |
17 | 14, 15, 16 | 3syl 18 | . 2 ∼ P6 ∼ 1c k SIk ∼ k |
18 | 1, 17 | syl5eqel 2437 | 1 k |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 358 wcel 1710 cvv 2859 ∼ ccompl 3205 cun 3207 cin 3208 1cc1c 4134 k cxpk 4174 P6 cp6 4178 kcimak 4179 SIk csik 4181 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-si 4083 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-ss 3259 df-nul 3551 df-sn 3741 df-pr 3742 df-opk 4058 df-1c 4136 df-xpk 4185 df-cnvk 4186 df-imak 4189 df-p6 4191 df-sik 4192 |
This theorem is referenced by: imakex 4300 pw1exg 4302 cokexg 4309 imagekexg 4311 uniexg 4316 intexg 4319 pwexg 4328 addcexg 4393 phiexg 4571 opexg 4587 proj1exg 4591 proj2exg 4592 imaexg 4746 coexg 4749 siexg 4752 |
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