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Mirrors > Home > MPE Home > Th. List > ex-res | Structured version Visualization version GIF version |
Description: Example for df-res 5681. 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 482 | . . . . 5 ⊢ ((𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} ∧ 𝐵 = {1, 2}) → 𝐹 = {⟨2, 6⟩, ⟨3, 9⟩}) | |
2 | df-pr 4626 | . . . . 5 ⊢ {⟨2, 6⟩, ⟨3, 9⟩} = ({⟨2, 6⟩} ∪ {⟨3, 9⟩}) | |
3 | 1, 2 | eqtrdi 2782 | . . . 4 ⊢ ((𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} ∧ 𝐵 = {1, 2}) → 𝐹 = ({⟨2, 6⟩} ∪ {⟨3, 9⟩})) |
4 | 3 | reseq1d 5973 | . . 3 ⊢ ((𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} ∧ 𝐵 = {1, 2}) → (𝐹 ↾ 𝐵) = (({⟨2, 6⟩} ∪ {⟨3, 9⟩}) ↾ 𝐵)) |
5 | resundir 5989 | . . 3 ⊢ (({⟨2, 6⟩} ∪ {⟨3, 9⟩}) ↾ 𝐵) = (({⟨2, 6⟩} ↾ 𝐵) ∪ ({⟨3, 9⟩} ↾ 𝐵)) | |
6 | 4, 5 | eqtrdi 2782 | . 2 ⊢ ((𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} ∧ 𝐵 = {1, 2}) → (𝐹 ↾ 𝐵) = (({⟨2, 6⟩} ↾ 𝐵) ∪ ({⟨3, 9⟩} ↾ 𝐵))) |
7 | 2re 12287 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
8 | 7 | elexi 3488 | . . . . . 6 ⊢ 2 ∈ V |
9 | 6re 12303 | . . . . . . 7 ⊢ 6 ∈ ℝ | |
10 | 9 | elexi 3488 | . . . . . 6 ⊢ 6 ∈ V |
11 | 8, 10 | relsnop 5798 | . . . . 5 ⊢ Rel {⟨2, 6⟩} |
12 | dmsnopss 6206 | . . . . . 6 ⊢ dom {⟨2, 6⟩} ⊆ {2} | |
13 | snsspr2 4813 | . . . . . . 7 ⊢ {2} ⊆ {1, 2} | |
14 | simpr 484 | . . . . . . 7 ⊢ ((𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} ∧ 𝐵 = {1, 2}) → 𝐵 = {1, 2}) | |
15 | 13, 14 | sseqtrrid 4030 | . . . . . 6 ⊢ ((𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} ∧ 𝐵 = {1, 2}) → {2} ⊆ 𝐵) |
16 | 12, 15 | sstrid 3988 | . . . . 5 ⊢ ((𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} ∧ 𝐵 = {1, 2}) → dom {⟨2, 6⟩} ⊆ 𝐵) |
17 | relssres 6015 | . . . . 5 ⊢ ((Rel {⟨2, 6⟩} ∧ dom {⟨2, 6⟩} ⊆ 𝐵) → ({⟨2, 6⟩} ↾ 𝐵) = {⟨2, 6⟩}) | |
18 | 11, 16, 17 | sylancr 586 | . . . 4 ⊢ ((𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} ∧ 𝐵 = {1, 2}) → ({⟨2, 6⟩} ↾ 𝐵) = {⟨2, 6⟩}) |
19 | 1re 11215 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
20 | 1lt3 12386 | . . . . . . . 8 ⊢ 1 < 3 | |
21 | 19, 20 | gtneii 11327 | . . . . . . 7 ⊢ 3 ≠ 1 |
22 | 2lt3 12385 | . . . . . . . 8 ⊢ 2 < 3 | |
23 | 7, 22 | gtneii 11327 | . . . . . . 7 ⊢ 3 ≠ 2 |
24 | 21, 23 | nelpri 4652 | . . . . . 6 ⊢ ¬ 3 ∈ {1, 2} |
25 | 14 | eleq2d 2813 | . . . . . 6 ⊢ ((𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} ∧ 𝐵 = {1, 2}) → (3 ∈ 𝐵 ↔ 3 ∈ {1, 2})) |
26 | 24, 25 | mtbiri 327 | . . . . 5 ⊢ ((𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} ∧ 𝐵 = {1, 2}) → ¬ 3 ∈ 𝐵) |
27 | ressnop0 7146 | . . . . 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 4159 | . . 3 ⊢ ((𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} ∧ 𝐵 = {1, 2}) → (({⟨2, 6⟩} ↾ 𝐵) ∪ ({⟨3, 9⟩} ↾ 𝐵)) = ({⟨2, 6⟩} ∪ ∅)) |
30 | un0 4385 | . . 3 ⊢ ({⟨2, 6⟩} ∪ ∅) = {⟨2, 6⟩} | |
31 | 29, 30 | eqtrdi 2782 | . 2 ⊢ ((𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} ∧ 𝐵 = {1, 2}) → (({⟨2, 6⟩} ↾ 𝐵) ∪ ({⟨3, 9⟩} ↾ 𝐵)) = {⟨2, 6⟩}) |
32 | 6, 31 | eqtrd 2766 | 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 395 = wceq 1533 ∈ wcel 2098 ∪ cun 3941 ⊆ wss 3943 ∅c0 4317 {csn 4623 {cpr 4625 ⟨cop 4629 dom cdm 5669 ↾ cres 5671 Rel wrel 5674 ℝcr 11108 1c1 11110 2c2 12268 3c3 12269 6c6 12272 9c9 12275 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 |
This theorem is referenced by: ex-ima 30200 |
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