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Mirrors > Home > NFE Home > Th. List > cnvuni | GIF version |
Description: The converse of a class union is the (indexed) union of the converses of its members. (Contributed by set.mm contributors, 11-Aug-2004.) |
Ref | Expression |
---|---|
cnvuni | ⊢ ◡∪A = ∪x ∈ A ◡x |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elcnv2 4890 | . . . 4 ⊢ (y ∈ ◡∪A ↔ ∃z∃w(y = 〈z, w〉 ∧ 〈w, z〉 ∈ ∪A)) | |
2 | eluni2 3895 | . . . . . . 7 ⊢ (〈w, z〉 ∈ ∪A ↔ ∃x ∈ A 〈w, z〉 ∈ x) | |
3 | 2 | anbi2i 675 | . . . . . 6 ⊢ ((y = 〈z, w〉 ∧ 〈w, z〉 ∈ ∪A) ↔ (y = 〈z, w〉 ∧ ∃x ∈ A 〈w, z〉 ∈ x)) |
4 | r19.42v 2765 | . . . . . 6 ⊢ (∃x ∈ A (y = 〈z, w〉 ∧ 〈w, z〉 ∈ x) ↔ (y = 〈z, w〉 ∧ ∃x ∈ A 〈w, z〉 ∈ x)) | |
5 | 3, 4 | bitr4i 243 | . . . . 5 ⊢ ((y = 〈z, w〉 ∧ 〈w, z〉 ∈ ∪A) ↔ ∃x ∈ A (y = 〈z, w〉 ∧ 〈w, z〉 ∈ x)) |
6 | 5 | 2exbii 1583 | . . . 4 ⊢ (∃z∃w(y = 〈z, w〉 ∧ 〈w, z〉 ∈ ∪A) ↔ ∃z∃w∃x ∈ A (y = 〈z, w〉 ∧ 〈w, z〉 ∈ x)) |
7 | elcnv2 4890 | . . . . . 6 ⊢ (y ∈ ◡x ↔ ∃z∃w(y = 〈z, w〉 ∧ 〈w, z〉 ∈ x)) | |
8 | 7 | rexbii 2639 | . . . . 5 ⊢ (∃x ∈ A y ∈ ◡x ↔ ∃x ∈ A ∃z∃w(y = 〈z, w〉 ∧ 〈w, z〉 ∈ x)) |
9 | rexcom4 2878 | . . . . 5 ⊢ (∃x ∈ A ∃z∃w(y = 〈z, w〉 ∧ 〈w, z〉 ∈ x) ↔ ∃z∃x ∈ A ∃w(y = 〈z, w〉 ∧ 〈w, z〉 ∈ x)) | |
10 | rexcom4 2878 | . . . . . 6 ⊢ (∃x ∈ A ∃w(y = 〈z, w〉 ∧ 〈w, z〉 ∈ x) ↔ ∃w∃x ∈ A (y = 〈z, w〉 ∧ 〈w, z〉 ∈ x)) | |
11 | 10 | exbii 1582 | . . . . 5 ⊢ (∃z∃x ∈ A ∃w(y = 〈z, w〉 ∧ 〈w, z〉 ∈ x) ↔ ∃z∃w∃x ∈ A (y = 〈z, w〉 ∧ 〈w, z〉 ∈ x)) |
12 | 8, 9, 11 | 3bitrri 263 | . . . 4 ⊢ (∃z∃w∃x ∈ A (y = 〈z, w〉 ∧ 〈w, z〉 ∈ x) ↔ ∃x ∈ A y ∈ ◡x) |
13 | 1, 6, 12 | 3bitri 262 | . . 3 ⊢ (y ∈ ◡∪A ↔ ∃x ∈ A y ∈ ◡x) |
14 | eliun 3973 | . . 3 ⊢ (y ∈ ∪x ∈ A ◡x ↔ ∃x ∈ A y ∈ ◡x) | |
15 | 13, 14 | bitr4i 243 | . 2 ⊢ (y ∈ ◡∪A ↔ y ∈ ∪x ∈ A ◡x) |
16 | 15 | eqriv 2350 | 1 ⊢ ◡∪A = ∪x ∈ A ◡x |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 358 ∃wex 1541 = wceq 1642 ∈ wcel 1710 ∃wrex 2615 ∪cuni 3891 ∪ciun 3969 〈cop 4561 ◡ccnv 4771 |
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-iun 3971 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-br 4640 df-cnv 4785 |
This theorem is referenced by: funcnvuni 5161 |
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