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Mirrors > Home > MPE Home > Th. List > ex-res | Structured version Visualization version GIF version |
Description: Example for df-res 5155. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.) |
Ref | Expression |
---|---|
ex-res | ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → (𝐹 ↾ 𝐵) = {〈2, 6〉}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 472 | . . . . 5 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → 𝐹 = {〈2, 6〉, 〈3, 9〉}) | |
2 | df-pr 4213 | . . . . 5 ⊢ {〈2, 6〉, 〈3, 9〉} = ({〈2, 6〉} ∪ {〈3, 9〉}) | |
3 | 1, 2 | syl6eq 2701 | . . . 4 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → 𝐹 = ({〈2, 6〉} ∪ {〈3, 9〉})) |
4 | 3 | reseq1d 5427 | . . 3 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → (𝐹 ↾ 𝐵) = (({〈2, 6〉} ∪ {〈3, 9〉}) ↾ 𝐵)) |
5 | resundir 5446 | . . 3 ⊢ (({〈2, 6〉} ∪ {〈3, 9〉}) ↾ 𝐵) = (({〈2, 6〉} ↾ 𝐵) ∪ ({〈3, 9〉} ↾ 𝐵)) | |
6 | 4, 5 | syl6eq 2701 | . 2 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → (𝐹 ↾ 𝐵) = (({〈2, 6〉} ↾ 𝐵) ∪ ({〈3, 9〉} ↾ 𝐵))) |
7 | 2re 11128 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
8 | 7 | elexi 3244 | . . . . . 6 ⊢ 2 ∈ V |
9 | 6re 11139 | . . . . . . 7 ⊢ 6 ∈ ℝ | |
10 | 9 | elexi 3244 | . . . . . 6 ⊢ 6 ∈ V |
11 | 8, 10 | relsnop 5261 | . . . . 5 ⊢ Rel {〈2, 6〉} |
12 | dmsnopss 5643 | . . . . . 6 ⊢ dom {〈2, 6〉} ⊆ {2} | |
13 | snsspr2 4378 | . . . . . . 7 ⊢ {2} ⊆ {1, 2} | |
14 | simpr 476 | . . . . . . 7 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → 𝐵 = {1, 2}) | |
15 | 13, 14 | syl5sseqr 3687 | . . . . . 6 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → {2} ⊆ 𝐵) |
16 | 12, 15 | syl5ss 3647 | . . . . 5 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → dom {〈2, 6〉} ⊆ 𝐵) |
17 | relssres 5472 | . . . . 5 ⊢ ((Rel {〈2, 6〉} ∧ dom {〈2, 6〉} ⊆ 𝐵) → ({〈2, 6〉} ↾ 𝐵) = {〈2, 6〉}) | |
18 | 11, 16, 17 | sylancr 696 | . . . 4 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → ({〈2, 6〉} ↾ 𝐵) = {〈2, 6〉}) |
19 | 1re 10077 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
20 | 1lt3 11234 | . . . . . . . 8 ⊢ 1 < 3 | |
21 | 19, 20 | gtneii 10187 | . . . . . . 7 ⊢ 3 ≠ 1 |
22 | 2lt3 11233 | . . . . . . . 8 ⊢ 2 < 3 | |
23 | 7, 22 | gtneii 10187 | . . . . . . 7 ⊢ 3 ≠ 2 |
24 | 21, 23 | nelpri 4234 | . . . . . 6 ⊢ ¬ 3 ∈ {1, 2} |
25 | 14 | eleq2d 2716 | . . . . . 6 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → (3 ∈ 𝐵 ↔ 3 ∈ {1, 2})) |
26 | 24, 25 | mtbiri 316 | . . . . 5 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → ¬ 3 ∈ 𝐵) |
27 | ressnop0 6460 | . . . . 5 ⊢ (¬ 3 ∈ 𝐵 → ({〈3, 9〉} ↾ 𝐵) = ∅) | |
28 | 26, 27 | syl 17 | . . . 4 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → ({〈3, 9〉} ↾ 𝐵) = ∅) |
29 | 18, 28 | uneq12d 3801 | . . 3 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → (({〈2, 6〉} ↾ 𝐵) ∪ ({〈3, 9〉} ↾ 𝐵)) = ({〈2, 6〉} ∪ ∅)) |
30 | un0 4000 | . . 3 ⊢ ({〈2, 6〉} ∪ ∅) = {〈2, 6〉} | |
31 | 29, 30 | syl6eq 2701 | . 2 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → (({〈2, 6〉} ↾ 𝐵) ∪ ({〈3, 9〉} ↾ 𝐵)) = {〈2, 6〉}) |
32 | 6, 31 | eqtrd 2685 | 1 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → (𝐹 ↾ 𝐵) = {〈2, 6〉}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ∪ cun 3605 ⊆ wss 3607 ∅c0 3948 {csn 4210 {cpr 4212 〈cop 4216 dom cdm 5143 ↾ cres 5145 Rel wrel 5148 ℝcr 9973 1c1 9975 2c2 11108 3c3 11109 6c6 11112 9c9 11115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-po 5064 df-so 5065 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 |
This theorem is referenced by: ex-ima 27429 |
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