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Mirrors > Home > NFE Home > Th. List > imagekeq | Unicode version |
Description: Equality theorem for image operation. (Contributed by SF, 12-Jan-2015.) |
Ref | Expression |
---|---|
imagekeq | Imagek Imagek |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sikeq 4241 | . . . . . . . 8 SIk SIk | |
2 | 1 | cnvkeqd 4217 | . . . . . . 7 k SIk k SIk |
3 | 2 | cokeq2d 4235 | . . . . . 6 Sk k k SIk Sk k k SIk |
4 | 3 | ins3keqd 4223 | . . . . 5 Ins3k Sk k k SIk Ins3k Sk k k SIk |
5 | 4 | symdifeq2d 3255 | . . . 4 Ins2k Sk Ins3k Sk k k SIk Ins2k Sk Ins3k Sk k k SIk |
6 | 5 | imakeq1d 4228 | . . 3 Ins2k Sk Ins3k Sk k k SIk k1 1 1c Ins2k Sk Ins3k Sk k k SIk k1 1 1c |
7 | 6 | difeq2d 3385 | . 2 k Ins2k Sk Ins3k Sk k k SIk k1 1 1c k Ins2k Sk Ins3k Sk k k SIk k1 1 1c |
8 | df-imagek 4194 | . 2 Imagek k Ins2k Sk Ins3k Sk k k SIk k1 1 1c | |
9 | df-imagek 4194 | . 2 Imagek k Ins2k Sk Ins3k Sk k k SIk k1 1 1c | |
10 | 7, 8, 9 | 3eqtr4g 2410 | 1 Imagek Imagek |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wceq 1642 cvv 2859 cdif 3206 csymdif 3209 1cc1c 4134 1 cpw1 4135 k cxpk 4174 kccnvk 4175 Ins2k cins2k 4176 Ins3k cins3k 4177 kcimak 4179 k ccomk 4180 SIk csik 4181 Imagekcimagek 4182 Sk cssetk 4183 |
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-un 3214 df-dif 3215 df-symdif 3216 df-cnvk 4186 df-ins3k 4188 df-imak 4189 df-cok 4190 df-sik 4192 df-imagek 4194 |
This theorem is referenced by: (None) |
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