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Mirrors > Home > NFE Home > Th. List > dfsset2 | GIF version |
Description: Express the S relationship via the set construction functors. (Contributed by SF, 7-Jan-2015.) |
Ref | Expression |
---|---|
dfsset2 | ⊢ S = ⋃1⋃1((((V ×k V) ×k V) ∩ ◡k ∼ (( Ins3k SIk SIk Sk ⊕ Ins2k ( Ins3k ( Sk ∘k SIk ◡kImagek((Imagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘11c) ∩ ( Nn ×k V)) ∪ ( Ik ∩ ( ∼ Nn ×k V)))) ∪ Ins2k (( Ins2k Sk ∩ Ins3k SIk ∼ (( Ins2k Sk ⊕ Ins3k ((◡kImagek((Imagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘11c) ∩ ( Nn ×k V)) ∪ ( Ik ∩ ( ∼ Nn ×k V))) ∘k Sk ) ∪ ({{0c}} ×k V))) “k ℘1℘11c)) “k ℘1℘11c))) “k ℘1℘1℘1℘11c)) “k Sk ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2862 | . . . 4 ⊢ x ∈ V | |
2 | vex 2862 | . . . 4 ⊢ y ∈ V | |
3 | opkelssetkg 4268 | . . . 4 ⊢ ((x ∈ V ∧ y ∈ V) → (⟪x, y⟫ ∈ Sk ↔ x ⊆ y)) | |
4 | 1, 2, 3 | mp2an 653 | . . 3 ⊢ (⟪x, y⟫ ∈ Sk ↔ x ⊆ y) |
5 | 4 | opabbii 4626 | . 2 ⊢ {〈x, y〉 ∣ ⟪x, y⟫ ∈ Sk } = {〈x, y〉 ∣ x ⊆ y} |
6 | setconslem4 4734 | . 2 ⊢ ⋃1⋃1((((V ×k V) ×k V) ∩ ◡k ∼ (( Ins3k SIk SIk Sk ⊕ Ins2k ( Ins3k ( Sk ∘k SIk ◡kImagek((Imagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘11c) ∩ ( Nn ×k V)) ∪ ( Ik ∩ ( ∼ Nn ×k V)))) ∪ Ins2k (( Ins2k Sk ∩ Ins3k SIk ∼ (( Ins2k Sk ⊕ Ins3k ((◡kImagek((Imagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘11c) ∩ ( Nn ×k V)) ∪ ( Ik ∩ ( ∼ Nn ×k V))) ∘k Sk ) ∪ ({{0c}} ×k V))) “k ℘1℘11c)) “k ℘1℘11c))) “k ℘1℘1℘1℘11c)) “k Sk ) = {〈x, y〉 ∣ ⟪x, y⟫ ∈ Sk } | |
7 | df-sset 4725 | . 2 ⊢ S = {〈x, y〉 ∣ x ⊆ y} | |
8 | 5, 6, 7 | 3eqtr4ri 2384 | 1 ⊢ S = ⋃1⋃1((((V ×k V) ×k V) ∩ ◡k ∼ (( Ins3k SIk SIk Sk ⊕ Ins2k ( Ins3k ( Sk ∘k SIk ◡kImagek((Imagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘11c) ∩ ( Nn ×k V)) ∪ ( Ik ∩ ( ∼ Nn ×k V)))) ∪ Ins2k (( Ins2k Sk ∩ Ins3k SIk ∼ (( Ins2k Sk ⊕ Ins3k ((◡kImagek((Imagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘11c) ∩ ( Nn ×k V)) ∪ ( Ik ∩ ( ∼ Nn ×k V))) ∘k Sk ) ∪ ({{0c}} ×k V))) “k ℘1℘11c)) “k ℘1℘11c))) “k ℘1℘1℘1℘11c)) “k Sk ) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 = wceq 1642 ∈ wcel 1710 Vcvv 2859 ∼ ccompl 3205 ∖ cdif 3206 ∪ cun 3207 ∩ cin 3208 ⊕ csymdif 3209 ⊆ wss 3257 {csn 3737 ⟪copk 4057 ⋃1cuni1 4133 1cc1c 4134 ℘1cpw1 4135 ×k cxpk 4174 ◡kccnvk 4175 Ins2k cins2k 4176 Ins3k cins3k 4177 “k cimak 4179 ∘k ccomk 4180 SIk csik 4181 Imagekcimagek 4182 Sk cssetk 4183 Ik cidk 4184 Nn cnnc 4373 0cc0c 4374 {copab 4622 S csset 4719 |
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-sset 4082 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-sbc 3047 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-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-addc 4378 df-nnc 4379 df-phi 4565 df-op 4566 df-opab 4623 df-sset 4725 |
This theorem is referenced by: ssetex 4744 |
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