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Mirrors > Home > NFE Home > Th. List > proj1eq | Unicode version |
Description: Equality theorem for first projection operator. (Contributed by SF, 2-Jan-2015.) |
Ref | Expression |
---|---|
proj1eq | Proj1 Proj1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imakeq2 4225 | . 2 kImagekImagek Ins3k ∼ Ins3k Sk Ins2k Sk k1 1 1c Ins2k Ins2k Sk Ins2k Ins3k Sk Ins3k SIk SIk Sk k1 1 1 1 1ck1 1 1c Nn k k ∼ Nn k k kImagekImagek Ins3k ∼ Ins3k Sk Ins2k Sk k1 1 1c Ins2k Ins2k Sk Ins2k Ins3k Sk Ins3k SIk SIk Sk k1 1 1 1 1ck1 1 1c Nn k k ∼ Nn k k | |
2 | dfproj12 4576 | . 2 Proj1 kImagekImagek Ins3k ∼ Ins3k Sk Ins2k Sk k1 1 1c Ins2k Ins2k Sk Ins2k Ins3k Sk Ins3k SIk SIk Sk k1 1 1 1 1ck1 1 1c Nn k k ∼ Nn k k | |
3 | dfproj12 4576 | . 2 Proj1 kImagekImagek Ins3k ∼ Ins3k Sk Ins2k Sk k1 1 1c Ins2k Ins2k Sk Ins2k Ins3k Sk Ins3k SIk SIk Sk k1 1 1 1 1ck1 1 1c Nn k k ∼ Nn k k | |
4 | 1, 2, 3 | 3eqtr4g 2410 | 1 Proj1 Proj1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wceq 1642 cvv 2859 ∼ ccompl 3205 cdif 3206 cun 3207 cin 3208 csymdif 3209 1cc1c 4134 1 cpw1 4135 k cxpk 4174 kccnvk 4175 Ins2k cins2k 4176 Ins3k cins3k 4177 kcimak 4179 SIk csik 4181 Imagekcimagek 4182 Sk cssetk 4183 k cidk 4184 Nn cnnc 4373 Proj1 cproj1 4563 |
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-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-symdif 3216 df-ss 3259 df-nul 3551 df-if 3663 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-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-addc 4378 df-phi 4565 df-proj1 4567 |
This theorem is referenced by: opth 4602 |
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