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Mirrors > Home > NFE Home > Th. List > cokeq1 | GIF version |
Description: Equality theorem for Kuratowski composition of two classes. (Contributed by SF, 12-Jan-2015.) |
Ref | Expression |
---|---|
cokeq1 | ⊢ (A = B → (A ∘k C) = (B ∘k C)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ins2keq 4218 | . . . 4 ⊢ (A = B → Ins2k A = Ins2k B) | |
2 | 1 | ineq1d 3456 | . . 3 ⊢ (A = B → ( Ins2k A ∩ Ins3k ◡kC) = ( Ins2k B ∩ Ins3k ◡kC)) |
3 | 2 | imakeq1d 4228 | . 2 ⊢ (A = B → (( Ins2k A ∩ Ins3k ◡kC) “k V) = (( Ins2k B ∩ Ins3k ◡kC) “k V)) |
4 | df-cok 4190 | . 2 ⊢ (A ∘k C) = (( Ins2k A ∩ Ins3k ◡kC) “k V) | |
5 | df-cok 4190 | . 2 ⊢ (B ∘k C) = (( Ins2k B ∩ Ins3k ◡kC) “k V) | |
6 | 3, 4, 5 | 3eqtr4g 2410 | 1 ⊢ (A = B → (A ∘k C) = (B ∘k C)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1642 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-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 |
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-rex 2620 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-ins2k 4187 df-imak 4189 df-cok 4190 |
This theorem is referenced by: cokeq1i 4232 cokeq1d 4234 |
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