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Mirrors > Home > ILE Home > Th. List > dminss | GIF version |
Description: An upper bound for intersection with a domain. Theorem 40 of [Suppes] p. 66, who calls it "somewhat surprising." (Contributed by NM, 11-Aug-2004.) |
Ref | Expression |
---|---|
dminss | ⊢ (dom 𝑅 ∩ 𝐴) ⊆ (◡𝑅 “ (𝑅 “ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.8a 1482 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)) | |
2 | 1 | ancoms 255 | . . . . . 6 ⊢ ((𝑥𝑅𝑦 ∧ 𝑥 ∈ 𝐴) → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)) |
3 | vex 2560 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
4 | 3 | elima2 4674 | . . . . . 6 ⊢ (𝑦 ∈ (𝑅 “ 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)) |
5 | 2, 4 | sylibr 137 | . . . . 5 ⊢ ((𝑥𝑅𝑦 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ (𝑅 “ 𝐴)) |
6 | simpl 102 | . . . . . 6 ⊢ ((𝑥𝑅𝑦 ∧ 𝑥 ∈ 𝐴) → 𝑥𝑅𝑦) | |
7 | vex 2560 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
8 | 3, 7 | brcnv 4518 | . . . . . 6 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦) |
9 | 6, 8 | sylibr 137 | . . . . 5 ⊢ ((𝑥𝑅𝑦 ∧ 𝑥 ∈ 𝐴) → 𝑦◡𝑅𝑥) |
10 | 5, 9 | jca 290 | . . . 4 ⊢ ((𝑥𝑅𝑦 ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ (𝑅 “ 𝐴) ∧ 𝑦◡𝑅𝑥)) |
11 | 10 | eximi 1491 | . . 3 ⊢ (∃𝑦(𝑥𝑅𝑦 ∧ 𝑥 ∈ 𝐴) → ∃𝑦(𝑦 ∈ (𝑅 “ 𝐴) ∧ 𝑦◡𝑅𝑥)) |
12 | 7 | eldm 4532 | . . . . 5 ⊢ (𝑥 ∈ dom 𝑅 ↔ ∃𝑦 𝑥𝑅𝑦) |
13 | 12 | anbi1i 431 | . . . 4 ⊢ ((𝑥 ∈ dom 𝑅 ∧ 𝑥 ∈ 𝐴) ↔ (∃𝑦 𝑥𝑅𝑦 ∧ 𝑥 ∈ 𝐴)) |
14 | elin 3126 | . . . 4 ⊢ (𝑥 ∈ (dom 𝑅 ∩ 𝐴) ↔ (𝑥 ∈ dom 𝑅 ∧ 𝑥 ∈ 𝐴)) | |
15 | 19.41v 1782 | . . . 4 ⊢ (∃𝑦(𝑥𝑅𝑦 ∧ 𝑥 ∈ 𝐴) ↔ (∃𝑦 𝑥𝑅𝑦 ∧ 𝑥 ∈ 𝐴)) | |
16 | 13, 14, 15 | 3bitr4i 201 | . . 3 ⊢ (𝑥 ∈ (dom 𝑅 ∩ 𝐴) ↔ ∃𝑦(𝑥𝑅𝑦 ∧ 𝑥 ∈ 𝐴)) |
17 | 7 | elima2 4674 | . . 3 ⊢ (𝑥 ∈ (◡𝑅 “ (𝑅 “ 𝐴)) ↔ ∃𝑦(𝑦 ∈ (𝑅 “ 𝐴) ∧ 𝑦◡𝑅𝑥)) |
18 | 11, 16, 17 | 3imtr4i 190 | . 2 ⊢ (𝑥 ∈ (dom 𝑅 ∩ 𝐴) → 𝑥 ∈ (◡𝑅 “ (𝑅 “ 𝐴))) |
19 | 18 | ssriv 2949 | 1 ⊢ (dom 𝑅 ∩ 𝐴) ⊆ (◡𝑅 “ (𝑅 “ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ∃wex 1381 ∈ wcel 1393 ∩ cin 2916 ⊆ wss 2917 class class class wbr 3764 ◡ccnv 4344 dom cdm 4345 “ cima 4348 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-xp 4351 df-cnv 4353 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 |
This theorem is referenced by: (None) |
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