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| Description: Example for df-res 5697. 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 4629 | . . . . 5 ⊢ {〈2, 6〉, 〈3, 9〉} = ({〈2, 6〉} ∪ {〈3, 9〉}) | |
| 3 | 1, 2 | eqtrdi 2793 | . . . 4 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → 𝐹 = ({〈2, 6〉} ∪ {〈3, 9〉})) | 
| 4 | 3 | reseq1d 5996 | . . 3 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → (𝐹 ↾ 𝐵) = (({〈2, 6〉} ∪ {〈3, 9〉}) ↾ 𝐵)) | 
| 5 | resundir 6012 | . . 3 ⊢ (({〈2, 6〉} ∪ {〈3, 9〉}) ↾ 𝐵) = (({〈2, 6〉} ↾ 𝐵) ∪ ({〈3, 9〉} ↾ 𝐵)) | |
| 6 | 4, 5 | eqtrdi 2793 | . 2 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → (𝐹 ↾ 𝐵) = (({〈2, 6〉} ↾ 𝐵) ∪ ({〈3, 9〉} ↾ 𝐵))) | 
| 7 | 2re 12340 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
| 8 | 7 | elexi 3503 | . . . . . 6 ⊢ 2 ∈ V | 
| 9 | 6re 12356 | . . . . . . 7 ⊢ 6 ∈ ℝ | |
| 10 | 9 | elexi 3503 | . . . . . 6 ⊢ 6 ∈ V | 
| 11 | 8, 10 | relsnop 5815 | . . . . 5 ⊢ Rel {〈2, 6〉} | 
| 12 | dmsnopss 6234 | . . . . . 6 ⊢ dom {〈2, 6〉} ⊆ {2} | |
| 13 | snsspr2 4815 | . . . . . . 7 ⊢ {2} ⊆ {1, 2} | |
| 14 | simpr 484 | . . . . . . 7 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → 𝐵 = {1, 2}) | |
| 15 | 13, 14 | sseqtrrid 4027 | . . . . . 6 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → {2} ⊆ 𝐵) | 
| 16 | 12, 15 | sstrid 3995 | . . . . 5 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → dom {〈2, 6〉} ⊆ 𝐵) | 
| 17 | relssres 6040 | . . . . 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 11261 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
| 20 | 1lt3 12439 | . . . . . . . 8 ⊢ 1 < 3 | |
| 21 | 19, 20 | gtneii 11373 | . . . . . . 7 ⊢ 3 ≠ 1 | 
| 22 | 2lt3 12438 | . . . . . . . 8 ⊢ 2 < 3 | |
| 23 | 7, 22 | gtneii 11373 | . . . . . . 7 ⊢ 3 ≠ 2 | 
| 24 | 21, 23 | nelpri 4655 | . . . . . 6 ⊢ ¬ 3 ∈ {1, 2} | 
| 25 | 14 | eleq2d 2827 | . . . . . 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 7173 | . . . . 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 4169 | . . 3 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → (({〈2, 6〉} ↾ 𝐵) ∪ ({〈3, 9〉} ↾ 𝐵)) = ({〈2, 6〉} ∪ ∅)) | 
| 30 | un0 4394 | . . 3 ⊢ ({〈2, 6〉} ∪ ∅) = {〈2, 6〉} | |
| 31 | 29, 30 | eqtrdi 2793 | . 2 ⊢ ((𝐹 = {〈2, 6〉, 〈3, 9〉} ∧ 𝐵 = {1, 2}) → (({〈2, 6〉} ↾ 𝐵) ∪ ({〈3, 9〉} ↾ 𝐵)) = {〈2, 6〉}) | 
| 32 | 6, 31 | eqtrd 2777 | 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 1540 ∈ wcel 2108 ∪ cun 3949 ⊆ wss 3951 ∅c0 4333 {csn 4626 {cpr 4628 〈cop 4632 dom cdm 5685 ↾ cres 5687 Rel wrel 5690 ℝcr 11154 1c1 11156 2c2 12321 3c3 12322 6c6 12325 9c9 12328 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 | 
| This theorem is referenced by: ex-ima 30461 | 
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