![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > dminss | Structured version Visualization version 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 2179 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)) | |
2 | 1 | ancoms 458 | . . . . . 6 ⊢ ((𝑥𝑅𝑦 ∧ 𝑥 ∈ 𝐴) → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)) |
3 | vex 3482 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
4 | 3 | elima2 6086 | . . . . . 6 ⊢ (𝑦 ∈ (𝑅 “ 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)) |
5 | 2, 4 | sylibr 234 | . . . . 5 ⊢ ((𝑥𝑅𝑦 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ (𝑅 “ 𝐴)) |
6 | simpl 482 | . . . . . 6 ⊢ ((𝑥𝑅𝑦 ∧ 𝑥 ∈ 𝐴) → 𝑥𝑅𝑦) | |
7 | vex 3482 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
8 | 3, 7 | brcnv 5896 | . . . . . 6 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦) |
9 | 6, 8 | sylibr 234 | . . . . 5 ⊢ ((𝑥𝑅𝑦 ∧ 𝑥 ∈ 𝐴) → 𝑦◡𝑅𝑥) |
10 | 5, 9 | jca 511 | . . . 4 ⊢ ((𝑥𝑅𝑦 ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ (𝑅 “ 𝐴) ∧ 𝑦◡𝑅𝑥)) |
11 | 10 | eximi 1832 | . . 3 ⊢ (∃𝑦(𝑥𝑅𝑦 ∧ 𝑥 ∈ 𝐴) → ∃𝑦(𝑦 ∈ (𝑅 “ 𝐴) ∧ 𝑦◡𝑅𝑥)) |
12 | 7 | eldm 5914 | . . . . 5 ⊢ (𝑥 ∈ dom 𝑅 ↔ ∃𝑦 𝑥𝑅𝑦) |
13 | 12 | anbi1i 624 | . . . 4 ⊢ ((𝑥 ∈ dom 𝑅 ∧ 𝑥 ∈ 𝐴) ↔ (∃𝑦 𝑥𝑅𝑦 ∧ 𝑥 ∈ 𝐴)) |
14 | elin 3979 | . . . 4 ⊢ (𝑥 ∈ (dom 𝑅 ∩ 𝐴) ↔ (𝑥 ∈ dom 𝑅 ∧ 𝑥 ∈ 𝐴)) | |
15 | 19.41v 1947 | . . . 4 ⊢ (∃𝑦(𝑥𝑅𝑦 ∧ 𝑥 ∈ 𝐴) ↔ (∃𝑦 𝑥𝑅𝑦 ∧ 𝑥 ∈ 𝐴)) | |
16 | 13, 14, 15 | 3bitr4i 303 | . . 3 ⊢ (𝑥 ∈ (dom 𝑅 ∩ 𝐴) ↔ ∃𝑦(𝑥𝑅𝑦 ∧ 𝑥 ∈ 𝐴)) |
17 | 7 | elima2 6086 | . . 3 ⊢ (𝑥 ∈ (◡𝑅 “ (𝑅 “ 𝐴)) ↔ ∃𝑦(𝑦 ∈ (𝑅 “ 𝐴) ∧ 𝑦◡𝑅𝑥)) |
18 | 11, 16, 17 | 3imtr4i 292 | . 2 ⊢ (𝑥 ∈ (dom 𝑅 ∩ 𝐴) → 𝑥 ∈ (◡𝑅 “ (𝑅 “ 𝐴))) |
19 | 18 | ssriv 3999 | 1 ⊢ (dom 𝑅 ∩ 𝐴) ⊆ (◡𝑅 “ (𝑅 “ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 ∃wex 1776 ∈ wcel 2106 ∩ cin 3962 ⊆ wss 3963 class class class wbr 5148 ◡ccnv 5688 dom cdm 5689 “ cima 5692 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 |
This theorem is referenced by: lmhmlsp 21066 cnclsi 23296 kgencn3 23582 kqsat 23755 kqcldsat 23757 cfilucfil 24588 elrspunidl 33436 |
Copyright terms: Public domain | W3C validator |