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Mirrors > Home > NFE Home > Th. List > clos1ex | GIF version |
Description: The closure of a set under a set is a set. (Contributed by SF, 11-Feb-2015.) |
Ref | Expression |
---|---|
clos1ex.1 | ⊢ S ∈ V |
clos1ex.2 | ⊢ R ∈ V |
Ref | Expression |
---|---|
clos1ex | ⊢ Clos1 (S, R) ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-clos1 5873 | . 2 ⊢ Clos1 (S, R) = ∩{a ∣ (S ⊆ a ∧ (R “ a) ⊆ a)} | |
2 | elin 3219 | . . . . . 6 ⊢ (a ∈ (( S “ {S}) ∩ Fix ( S ∘ ImageR)) ↔ (a ∈ ( S “ {S}) ∧ a ∈ Fix ( S ∘ ImageR))) | |
3 | elimasn 5019 | . . . . . . . 8 ⊢ (a ∈ ( S “ {S}) ↔ 〈S, a〉 ∈ S ) | |
4 | df-br 4640 | . . . . . . . 8 ⊢ (S S a ↔ 〈S, a〉 ∈ S ) | |
5 | clos1ex.1 | . . . . . . . . 9 ⊢ S ∈ V | |
6 | vex 2862 | . . . . . . . . 9 ⊢ a ∈ V | |
7 | 5, 6 | brsset 4758 | . . . . . . . 8 ⊢ (S S a ↔ S ⊆ a) |
8 | 3, 4, 7 | 3bitr2i 264 | . . . . . . 7 ⊢ (a ∈ ( S “ {S}) ↔ S ⊆ a) |
9 | elfix 5787 | . . . . . . . 8 ⊢ (a ∈ Fix ( S ∘ ImageR) ↔ a( S ∘ ImageR)a) | |
10 | brco 4883 | . . . . . . . . 9 ⊢ (a( S ∘ ImageR)a ↔ ∃b(aImageRb ∧ b S a)) | |
11 | vex 2862 | . . . . . . . . . . . . 13 ⊢ b ∈ V | |
12 | 6, 11 | brimage 5793 | . . . . . . . . . . . 12 ⊢ (aImageRb ↔ b = (R “ a)) |
13 | 12 | anbi1i 676 | . . . . . . . . . . 11 ⊢ ((aImageRb ∧ b S a) ↔ (b = (R “ a) ∧ b S a)) |
14 | 13 | exbii 1582 | . . . . . . . . . 10 ⊢ (∃b(aImageRb ∧ b S a) ↔ ∃b(b = (R “ a) ∧ b S a)) |
15 | clos1ex.2 | . . . . . . . . . . . 12 ⊢ R ∈ V | |
16 | 15, 6 | imaex 4747 | . . . . . . . . . . 11 ⊢ (R “ a) ∈ V |
17 | breq1 4642 | . . . . . . . . . . . 12 ⊢ (b = (R “ a) → (b S a ↔ (R “ a) S a)) | |
18 | 16, 6 | brsset 4758 | . . . . . . . . . . . 12 ⊢ ((R “ a) S a ↔ (R “ a) ⊆ a) |
19 | 17, 18 | syl6bb 252 | . . . . . . . . . . 11 ⊢ (b = (R “ a) → (b S a ↔ (R “ a) ⊆ a)) |
20 | 16, 19 | ceqsexv 2894 | . . . . . . . . . 10 ⊢ (∃b(b = (R “ a) ∧ b S a) ↔ (R “ a) ⊆ a) |
21 | 14, 20 | bitri 240 | . . . . . . . . 9 ⊢ (∃b(aImageRb ∧ b S a) ↔ (R “ a) ⊆ a) |
22 | 10, 21 | bitri 240 | . . . . . . . 8 ⊢ (a( S ∘ ImageR)a ↔ (R “ a) ⊆ a) |
23 | 9, 22 | bitri 240 | . . . . . . 7 ⊢ (a ∈ Fix ( S ∘ ImageR) ↔ (R “ a) ⊆ a) |
24 | 8, 23 | anbi12i 678 | . . . . . 6 ⊢ ((a ∈ ( S “ {S}) ∧ a ∈ Fix ( S ∘ ImageR)) ↔ (S ⊆ a ∧ (R “ a) ⊆ a)) |
25 | 2, 24 | bitri 240 | . . . . 5 ⊢ (a ∈ (( S “ {S}) ∩ Fix ( S ∘ ImageR)) ↔ (S ⊆ a ∧ (R “ a) ⊆ a)) |
26 | 25 | abbi2i 2464 | . . . 4 ⊢ (( S “ {S}) ∩ Fix ( S ∘ ImageR)) = {a ∣ (S ⊆ a ∧ (R “ a) ⊆ a)} |
27 | ssetex 4744 | . . . . . 6 ⊢ S ∈ V | |
28 | snex 4111 | . . . . . 6 ⊢ {S} ∈ V | |
29 | 27, 28 | imaex 4747 | . . . . 5 ⊢ ( S “ {S}) ∈ V |
30 | 15 | imageex 5801 | . . . . . . 7 ⊢ ImageR ∈ V |
31 | 27, 30 | coex 4750 | . . . . . 6 ⊢ ( S ∘ ImageR) ∈ V |
32 | 31 | fixex 5789 | . . . . 5 ⊢ Fix ( S ∘ ImageR) ∈ V |
33 | 29, 32 | inex 4105 | . . . 4 ⊢ (( S “ {S}) ∩ Fix ( S ∘ ImageR)) ∈ V |
34 | 26, 33 | eqeltrri 2424 | . . 3 ⊢ {a ∣ (S ⊆ a ∧ (R “ a) ⊆ a)} ∈ V |
35 | 34 | intex 4320 | . 2 ⊢ ∩{a ∣ (S ⊆ a ∧ (R “ a) ⊆ a)} ∈ V |
36 | 1, 35 | eqeltri 2423 | 1 ⊢ Clos1 (S, R) ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 358 ∃wex 1541 = wceq 1642 ∈ wcel 1710 {cab 2339 Vcvv 2859 ∩ cin 3208 ⊆ wss 3257 {csn 3737 ∩cint 3926 〈cop 4561 class class class wbr 4639 S csset 4719 ∘ ccom 4721 “ cima 4722 Fix cfix 5739 Imagecimage 5753 Clos1 cclos1 5872 |
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-13 1712 ax-14 1714 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-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 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-pss 3261 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-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-1st 4723 df-swap 4724 df-sset 4725 df-co 4726 df-ima 4727 df-si 4728 df-id 4767 df-xp 4784 df-cnv 4785 df-rn 4786 df-dm 4787 df-res 4788 df-2nd 4797 df-txp 5736 df-fix 5740 df-ins2 5750 df-ins3 5752 df-image 5754 df-clos1 5873 |
This theorem is referenced by: clos1exg 5877 clos1induct 5880 clos1basesuc 5882 sbthlem1 6203 spacval 6282 fnspac 6283 nchoicelem11 6299 nchoicelem16 6304 |
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