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| Description: Example for df-res 5696. 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 4628 | . . . . 5 ⊢ {〈2, 6〉, 〈3, 9〉} = ({〈2, 6〉} ∪ {〈3, 9〉}) | |
| 3 | 1, 2 | eqtrdi 2792 | . . . 4 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → 𝐹 = ({〈2, 6〉} ∪ {〈3, 9〉})) | 
| 4 | 3 | reseq1d 5995 | . . 3 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → (𝐹 ↾ 𝐵) = (({〈2, 6〉} ∪ {〈3, 9〉}) ↾ 𝐵)) | 
| 5 | resundir 6011 | . . 3 ⊢ (({〈2, 6〉} ∪ {〈3, 9〉}) ↾ 𝐵) = (({〈2, 6〉} ↾ 𝐵) ∪ ({〈3, 9〉} ↾ 𝐵)) | |
| 6 | 4, 5 | eqtrdi 2792 | . 2 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → (𝐹 ↾ 𝐵) = (({〈2, 6〉} ↾ 𝐵) ∪ ({〈3, 9〉} ↾ 𝐵))) | 
| 7 | 2re 12341 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
| 8 | 7 | elexi 3502 | . . . . . 6 ⊢ 2 ∈ V | 
| 9 | 6re 12357 | . . . . . . 7 ⊢ 6 ∈ ℝ | |
| 10 | 9 | elexi 3502 | . . . . . 6 ⊢ 6 ∈ V | 
| 11 | 8, 10 | relsnop 5814 | . . . . 5 ⊢ Rel {〈2, 6〉} | 
| 12 | dmsnopss 6233 | . . . . . 6 ⊢ dom {〈2, 6〉} ⊆ {2} | |
| 13 | snsspr2 4814 | . . . . . . 7 ⊢ {2} ⊆ {1, 2} | |
| 14 | simpr 484 | . . . . . . 7 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → 𝐵 = {1, 2}) | |
| 15 | 13, 14 | sseqtrrid 4026 | . . . . . 6 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → {2} ⊆ 𝐵) | 
| 16 | 12, 15 | sstrid 3994 | . . . . 5 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → dom {〈2, 6〉} ⊆ 𝐵) | 
| 17 | relssres 6039 | . . . . 5 ⊢ ((Rel {〈2, 6〉} ∧ dom {〈2, 6〉} ⊆ 𝐵) → ({〈2, 6〉} ↾ 𝐵) = {〈2, 6〉}) | |
| 18 | 11, 16, 17 | sylancr 587 | . . . 4 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → ({〈2, 6〉} ↾ 𝐵) = {〈2, 6〉}) | 
| 19 | 1re 11262 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
| 20 | 1lt3 12440 | . . . . . . . 8 ⊢ 1 < 3 | |
| 21 | 19, 20 | gtneii 11374 | . . . . . . 7 ⊢ 3 ≠ 1 | 
| 22 | 2lt3 12439 | . . . . . . . 8 ⊢ 2 < 3 | |
| 23 | 7, 22 | gtneii 11374 | . . . . . . 7 ⊢ 3 ≠ 2 | 
| 24 | 21, 23 | nelpri 4654 | . . . . . 6 ⊢ ¬ 3 ∈ {1, 2} | 
| 25 | 14 | eleq2d 2826 | . . . . . 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 7172 | . . . . 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 4168 | . . 3 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → (({〈2, 6〉} ↾ 𝐵) ∪ ({〈3, 9〉} ↾ 𝐵)) = ({〈2, 6〉} ∪ ∅)) | 
| 30 | un0 4393 | . . 3 ⊢ ({〈2, 6〉} ∪ ∅) = {〈2, 6〉} | |
| 31 | 29, 30 | eqtrdi 2792 | . 2 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → (({〈2, 6〉} ↾ 𝐵) ∪ ({〈3, 9〉} ↾ 𝐵)) = {〈2, 6〉}) | 
| 32 | 6, 31 | eqtrd 2776 | 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 1539 ∈ wcel 2107 ∪ cun 3948 ⊆ wss 3950 ∅c0 4332 {csn 4625 {cpr 4627 〈cop 4631 dom cdm 5684 ↾ cres 5686 Rel wrel 5689 ℝcr 11155 1c1 11157 2c2 12322 3c3 12323 6c6 12326 9c9 12329 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-po 5591 df-so 5592 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 | 
| This theorem is referenced by: ex-ima 30462 | 
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