![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 6091 | . . 3 ⊢ (Rel 𝐴 → (𝑦 ∈ dom 𝐴 ↔ ∃𝑧 ∈ 𝐴 (1st ‘𝑧) = 𝑦)) | |
2 | vex 2692 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
3 | 1stexg 6073 | . . . . . . 7 ⊢ (𝑥 ∈ V → (1st ‘𝑥) ∈ V) | |
4 | 2, 3 | ax-mp 5 | . . . . . 6 ⊢ (1st ‘𝑥) ∈ V |
5 | eqid 2140 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥)) | |
6 | 4, 5 | fnmpti 5259 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥)) Fn 𝐴 |
7 | fvelrnb 5477 | . . . . 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 5 | . . . 4 ⊢ (𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥)) ↔ ∃𝑧 ∈ 𝐴 ((𝑥 ∈ 𝐴 ↦ (1st ‘𝑥))‘𝑧) = 𝑦) |
9 | fveq2 5429 | . . . . . . . 8 ⊢ (𝑥 = 𝑧 → (1st ‘𝑥) = (1st ‘𝑧)) | |
10 | vex 2692 | . . . . . . . . 9 ⊢ 𝑧 ∈ V | |
11 | 1stexg 6073 | . . . . . . . . 9 ⊢ (𝑧 ∈ V → (1st ‘𝑧) ∈ V) | |
12 | 10, 11 | ax-mp 5 | . . . . . . . 8 ⊢ (1st ‘𝑧) ∈ V |
13 | 9, 5, 12 | fvmpt 5506 | . . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ↦ (1st ‘𝑥))‘𝑧) = (1st ‘𝑧)) |
14 | 13 | eqeq1d 2149 | . . . . . 6 ⊢ (𝑧 ∈ 𝐴 → (((𝑥 ∈ 𝐴 ↦ (1st ‘𝑥))‘𝑧) = 𝑦 ↔ (1st ‘𝑧) = 𝑦)) |
15 | 14 | rexbiia 2453 | . . . . 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 192 | . . 3 ⊢ (Rel 𝐴 → (∃𝑧 ∈ 𝐴 (1st ‘𝑧) = 𝑦 ↔ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥)))) |
18 | 1, 17 | bitrd 187 | . 2 ⊢ (Rel 𝐴 → (𝑦 ∈ dom 𝐴 ↔ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥)))) |
19 | 18 | eqrdv 2138 | 1 ⊢ (Rel 𝐴 → dom 𝐴 = ran (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1332 ∈ wcel 1481 ∃wrex 2418 Vcvv 2689 ↦ cmpt 3997 dom cdm 4547 ran crn 4548 Rel wrel 4552 Fn wfn 5126 ‘cfv 5131 1st c1st 6044 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-fo 5137 df-fv 5139 df-1st 6046 df-2nd 6047 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |