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Mirrors > Home > NFE Home > Th. List > cokexg | GIF version |
Description: The Kuratowski composition of two sets is a set. (Contributed by SF, 14-Jan-2015.) |
Ref | Expression |
---|---|
cokexg | ⊢ ((A ∈ V ∧ B ∈ W) → (A ∘k B) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-cok 4190 | . 2 ⊢ (A ∘k B) = (( Ins2k A ∩ Ins3k ◡kB) “k V) | |
2 | ins2kexg 4305 | . . . 4 ⊢ (A ∈ V → Ins2k A ∈ V) | |
3 | cnvkexg 4286 | . . . . 5 ⊢ (B ∈ W → ◡kB ∈ V) | |
4 | ins3kexg 4306 | . . . . 5 ⊢ (◡kB ∈ V → Ins3k ◡kB ∈ V) | |
5 | 3, 4 | syl 15 | . . . 4 ⊢ (B ∈ W → Ins3k ◡kB ∈ V) |
6 | inexg 4100 | . . . 4 ⊢ (( Ins2k A ∈ V ∧ Ins3k ◡kB ∈ V) → ( Ins2k A ∩ Ins3k ◡kB) ∈ V) | |
7 | 2, 5, 6 | syl2an 463 | . . 3 ⊢ ((A ∈ V ∧ B ∈ W) → ( Ins2k A ∩ Ins3k ◡kB) ∈ V) |
8 | vvex 4109 | . . 3 ⊢ V ∈ V | |
9 | imakexg 4299 | . . 3 ⊢ ((( Ins2k A ∩ Ins3k ◡kB) ∈ V ∧ V ∈ V) → (( Ins2k A ∩ Ins3k ◡kB) “k V) ∈ V) | |
10 | 7, 8, 9 | sylancl 643 | . 2 ⊢ ((A ∈ V ∧ B ∈ W) → (( Ins2k A ∩ Ins3k ◡kB) “k V) ∈ V) |
11 | 1, 10 | syl5eqel 2437 | 1 ⊢ ((A ∈ V ∧ B ∈ W) → (A ∘k B) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 ∈ wcel 1710 Vcvv 2859 ∩ cin 3208 ◡kccnvk 4175 Ins2k cins2k 4176 Ins3k cins3k 4177 “k cimak 4179 ∘k ccomk 4180 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 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-ins2 4084 ax-ins3 4085 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-pw 3724 df-sn 3741 df-pr 3742 df-opk 4058 df-1c 4136 df-pw1 4137 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 |
This theorem is referenced by: cokex 4310 imagekexg 4311 coexg 4749 |
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