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Mirrors > Home > MPE Home > Th. List > ex-res | Structured version Visualization version GIF version |
Description: Example for df-res 5050. 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 4128 | . . . . 5 ⊢ {〈2, 6〉, 〈3, 9〉} = ({〈2, 6〉} ∪ {〈3, 9〉}) | |
3 | 1, 2 | syl6eq 2660 | . . . 4 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → 𝐹 = ({〈2, 6〉} ∪ {〈3, 9〉})) |
4 | 3 | reseq1d 5316 | . . 3 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → (𝐹 ↾ 𝐵) = (({〈2, 6〉} ∪ {〈3, 9〉}) ↾ 𝐵)) |
5 | resundir 5331 | . . 3 ⊢ (({〈2, 6〉} ∪ {〈3, 9〉}) ↾ 𝐵) = (({〈2, 6〉} ↾ 𝐵) ∪ ({〈3, 9〉} ↾ 𝐵)) | |
6 | 4, 5 | syl6eq 2660 | . 2 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → (𝐹 ↾ 𝐵) = (({〈2, 6〉} ↾ 𝐵) ∪ ({〈3, 9〉} ↾ 𝐵))) |
7 | 2re 10967 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
8 | 7 | elexi 3186 | . . . . . 6 ⊢ 2 ∈ V |
9 | 6re 10978 | . . . . . . 7 ⊢ 6 ∈ ℝ | |
10 | 9 | elexi 3186 | . . . . . 6 ⊢ 6 ∈ V |
11 | 8, 10 | relsnop 5147 | . . . . 5 ⊢ Rel {〈2, 6〉} |
12 | dmsnopss 5525 | . . . . . 6 ⊢ dom {〈2, 6〉} ⊆ {2} | |
13 | snsspr2 4286 | . . . . . . 7 ⊢ {2} ⊆ {1, 2} | |
14 | simpr 476 | . . . . . . 7 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → 𝐵 = {1, 2}) | |
15 | 13, 14 | syl5sseqr 3617 | . . . . . 6 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → {2} ⊆ 𝐵) |
16 | 12, 15 | syl5ss 3579 | . . . . 5 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → dom {〈2, 6〉} ⊆ 𝐵) |
17 | relssres 5357 | . . . . 5 ⊢ ((Rel {〈2, 6〉} ∧ dom {〈2, 6〉} ⊆ 𝐵) → ({〈2, 6〉} ↾ 𝐵) = {〈2, 6〉}) | |
18 | 11, 16, 17 | sylancr 694 | . . . 4 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → ({〈2, 6〉} ↾ 𝐵) = {〈2, 6〉}) |
19 | 1re 9918 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
20 | 1lt3 11073 | . . . . . . . 8 ⊢ 1 < 3 | |
21 | 19, 20 | gtneii 10028 | . . . . . . 7 ⊢ 3 ≠ 1 |
22 | 2lt3 11072 | . . . . . . . 8 ⊢ 2 < 3 | |
23 | 7, 22 | gtneii 10028 | . . . . . . 7 ⊢ 3 ≠ 2 |
24 | 21, 23 | nelpri 4149 | . . . . . 6 ⊢ ¬ 3 ∈ {1, 2} |
25 | 14 | eleq2d 2673 | . . . . . 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 6325 | . . . . 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 3730 | . . 3 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → (({〈2, 6〉} ↾ 𝐵) ∪ ({〈3, 9〉} ↾ 𝐵)) = ({〈2, 6〉} ∪ ∅)) |
30 | un0 3919 | . . 3 ⊢ ({〈2, 6〉} ∪ ∅) = {〈2, 6〉} | |
31 | 29, 30 | syl6eq 2660 | . 2 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → (({〈2, 6〉} ↾ 𝐵) ∪ ({〈3, 9〉} ↾ 𝐵)) = {〈2, 6〉}) |
32 | 6, 31 | eqtrd 2644 | 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 1475 ∈ wcel 1977 ∪ cun 3538 ⊆ wss 3540 ∅c0 3874 {csn 4125 {cpr 4127 〈cop 4131 dom cdm 5038 ↾ cres 5040 Rel wrel 5043 ℝcr 9814 1c1 9816 2c2 10947 3c3 10948 6c6 10951 9c9 10954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 |
This theorem is referenced by: ex-ima 26691 |
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