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Mirrors > Home > MPE Home > Th. List > ex-res | Structured version Visualization version GIF version |
Description: Example for df-res 5561. 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 485 | . . . . 5 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → 𝐹 = {〈2, 6〉, 〈3, 9〉}) | |
2 | df-pr 4563 | . . . . 5 ⊢ {〈2, 6〉, 〈3, 9〉} = ({〈2, 6〉} ∪ {〈3, 9〉}) | |
3 | 1, 2 | syl6eq 2872 | . . . 4 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → 𝐹 = ({〈2, 6〉} ∪ {〈3, 9〉})) |
4 | 3 | reseq1d 5846 | . . 3 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → (𝐹 ↾ 𝐵) = (({〈2, 6〉} ∪ {〈3, 9〉}) ↾ 𝐵)) |
5 | resundir 5862 | . . 3 ⊢ (({〈2, 6〉} ∪ {〈3, 9〉}) ↾ 𝐵) = (({〈2, 6〉} ↾ 𝐵) ∪ ({〈3, 9〉} ↾ 𝐵)) | |
6 | 4, 5 | syl6eq 2872 | . 2 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → (𝐹 ↾ 𝐵) = (({〈2, 6〉} ↾ 𝐵) ∪ ({〈3, 9〉} ↾ 𝐵))) |
7 | 2re 11705 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
8 | 7 | elexi 3513 | . . . . . 6 ⊢ 2 ∈ V |
9 | 6re 11721 | . . . . . . 7 ⊢ 6 ∈ ℝ | |
10 | 9 | elexi 3513 | . . . . . 6 ⊢ 6 ∈ V |
11 | 8, 10 | relsnop 5672 | . . . . 5 ⊢ Rel {〈2, 6〉} |
12 | dmsnopss 6065 | . . . . . 6 ⊢ dom {〈2, 6〉} ⊆ {2} | |
13 | snsspr2 4741 | . . . . . . 7 ⊢ {2} ⊆ {1, 2} | |
14 | simpr 487 | . . . . . . 7 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → 𝐵 = {1, 2}) | |
15 | 13, 14 | sseqtrrid 4019 | . . . . . 6 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → {2} ⊆ 𝐵) |
16 | 12, 15 | sstrid 3977 | . . . . 5 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → dom {〈2, 6〉} ⊆ 𝐵) |
17 | relssres 5887 | . . . . 5 ⊢ ((Rel {〈2, 6〉} ∧ dom {〈2, 6〉} ⊆ 𝐵) → ({〈2, 6〉} ↾ 𝐵) = {〈2, 6〉}) | |
18 | 11, 16, 17 | sylancr 589 | . . . 4 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → ({〈2, 6〉} ↾ 𝐵) = {〈2, 6〉}) |
19 | 1re 10635 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
20 | 1lt3 11804 | . . . . . . . 8 ⊢ 1 < 3 | |
21 | 19, 20 | gtneii 10746 | . . . . . . 7 ⊢ 3 ≠ 1 |
22 | 2lt3 11803 | . . . . . . . 8 ⊢ 2 < 3 | |
23 | 7, 22 | gtneii 10746 | . . . . . . 7 ⊢ 3 ≠ 2 |
24 | 21, 23 | nelpri 4587 | . . . . . 6 ⊢ ¬ 3 ∈ {1, 2} |
25 | 14 | eleq2d 2898 | . . . . . 6 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → (3 ∈ 𝐵 ↔ 3 ∈ {1, 2})) |
26 | 24, 25 | mtbiri 329 | . . . . 5 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → ¬ 3 ∈ 𝐵) |
27 | ressnop0 6909 | . . . . 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 4139 | . . 3 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → (({〈2, 6〉} ↾ 𝐵) ∪ ({〈3, 9〉} ↾ 𝐵)) = ({〈2, 6〉} ∪ ∅)) |
30 | un0 4343 | . . 3 ⊢ ({〈2, 6〉} ∪ ∅) = {〈2, 6〉} | |
31 | 29, 30 | syl6eq 2872 | . 2 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → (({〈2, 6〉} ↾ 𝐵) ∪ ({〈3, 9〉} ↾ 𝐵)) = {〈2, 6〉}) |
32 | 6, 31 | eqtrd 2856 | 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 398 = wceq 1533 ∈ wcel 2110 ∪ cun 3933 ⊆ wss 3935 ∅c0 4290 {csn 4560 {cpr 4562 〈cop 4566 dom cdm 5549 ↾ cres 5551 Rel wrel 5554 ℝcr 10530 1c1 10532 2c2 11686 3c3 11687 6c6 11690 9c9 11693 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 |
This theorem is referenced by: ex-ima 28215 |
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