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Mirrors > Home > ILE Home > Th. List > reldm | GIF version |
Description: An expression for the domain of a relation. (Contributed by NM, 22-Sep-2013.) |
Ref | Expression |
---|---|
reldm | ⊢ (Rel 𝐴 → dom 𝐴 = ran (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | releldm2 5831 | . . 3 ⊢ (Rel 𝐴 → (𝑦 ∈ dom 𝐴 ↔ ∃𝑧 ∈ 𝐴 (1st ‘𝑧) = 𝑦)) | |
2 | vex 2604 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
3 | 1stexg 5814 | . . . . . . 7 ⊢ (𝑥 ∈ V → (1st ‘𝑥) ∈ V) | |
4 | 2, 3 | ax-mp 7 | . . . . . 6 ⊢ (1st ‘𝑥) ∈ V |
5 | eqid 2081 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥)) | |
6 | 4, 5 | fnmpti 5047 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥)) Fn 𝐴 |
7 | fvelrnb 5242 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↦ (1st ‘𝑥)) Fn 𝐴 → (𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥)) ↔ ∃𝑧 ∈ 𝐴 ((𝑥 ∈ 𝐴 ↦ (1st ‘𝑥))‘𝑧) = 𝑦)) | |
8 | 6, 7 | ax-mp 7 | . . . 4 ⊢ (𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥)) ↔ ∃𝑧 ∈ 𝐴 ((𝑥 ∈ 𝐴 ↦ (1st ‘𝑥))‘𝑧) = 𝑦) |
9 | fveq2 5198 | . . . . . . . 8 ⊢ (𝑥 = 𝑧 → (1st ‘𝑥) = (1st ‘𝑧)) | |
10 | vex 2604 | . . . . . . . . 9 ⊢ 𝑧 ∈ V | |
11 | 1stexg 5814 | . . . . . . . . 9 ⊢ (𝑧 ∈ V → (1st ‘𝑧) ∈ V) | |
12 | 10, 11 | ax-mp 7 | . . . . . . . 8 ⊢ (1st ‘𝑧) ∈ V |
13 | 9, 5, 12 | fvmpt 5270 | . . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ↦ (1st ‘𝑥))‘𝑧) = (1st ‘𝑧)) |
14 | 13 | eqeq1d 2089 | . . . . . 6 ⊢ (𝑧 ∈ 𝐴 → (((𝑥 ∈ 𝐴 ↦ (1st ‘𝑥))‘𝑧) = 𝑦 ↔ (1st ‘𝑧) = 𝑦)) |
15 | 14 | rexbiia 2381 | . . . . 5 ⊢ (∃𝑧 ∈ 𝐴 ((𝑥 ∈ 𝐴 ↦ (1st ‘𝑥))‘𝑧) = 𝑦 ↔ ∃𝑧 ∈ 𝐴 (1st ‘𝑧) = 𝑦) |
16 | 15 | a1i 9 | . . . 4 ⊢ (Rel 𝐴 → (∃𝑧 ∈ 𝐴 ((𝑥 ∈ 𝐴 ↦ (1st ‘𝑥))‘𝑧) = 𝑦 ↔ ∃𝑧 ∈ 𝐴 (1st ‘𝑧) = 𝑦)) |
17 | 8, 16 | syl5rbb 191 | . . 3 ⊢ (Rel 𝐴 → (∃𝑧 ∈ 𝐴 (1st ‘𝑧) = 𝑦 ↔ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥)))) |
18 | 1, 17 | bitrd 186 | . 2 ⊢ (Rel 𝐴 → (𝑦 ∈ dom 𝐴 ↔ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥)))) |
19 | 18 | eqrdv 2079 | 1 ⊢ (Rel 𝐴 → dom 𝐴 = ran (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 = wceq 1284 ∈ wcel 1433 ∃wrex 2349 Vcvv 2601 ↦ cmpt 3839 dom cdm 4363 ran crn 4364 Rel wrel 4368 Fn wfn 4917 ‘cfv 4922 1st c1st 5785 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-sbc 2816 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-fo 4928 df-fv 4930 df-1st 5787 df-2nd 5788 |
This theorem is referenced by: (None) |
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