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Mirrors > Home > MPE Home > Th. List > ex-res | Structured version Visualization version GIF version |
Description: Example for df-res 5650. 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 484 | . . . . 5 ⊢ ((𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} ∧ 𝐵 = {1, 2}) → 𝐹 = {⟨2, 6⟩, ⟨3, 9⟩}) | |
2 | df-pr 4594 | . . . . 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 5941 | . . 3 ⊢ ((𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} ∧ 𝐵 = {1, 2}) → (𝐹 ↾ 𝐵) = (({⟨2, 6⟩} ∪ {⟨3, 9⟩}) ↾ 𝐵)) |
5 | resundir 5957 | . . 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 12234 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
8 | 7 | elexi 3467 | . . . . . 6 ⊢ 2 ∈ V |
9 | 6re 12250 | . . . . . . 7 ⊢ 6 ∈ ℝ | |
10 | 9 | elexi 3467 | . . . . . 6 ⊢ 6 ∈ V |
11 | 8, 10 | relsnop 5766 | . . . . 5 ⊢ Rel {⟨2, 6⟩} |
12 | dmsnopss 6171 | . . . . . 6 ⊢ dom {⟨2, 6⟩} ⊆ {2} | |
13 | snsspr2 4780 | . . . . . . 7 ⊢ {2} ⊆ {1, 2} | |
14 | simpr 486 | . . . . . . 7 ⊢ ((𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} ∧ 𝐵 = {1, 2}) → 𝐵 = {1, 2}) | |
15 | 13, 14 | sseqtrrid 4002 | . . . . . 6 ⊢ ((𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} ∧ 𝐵 = {1, 2}) → {2} ⊆ 𝐵) |
16 | 12, 15 | sstrid 3960 | . . . . 5 ⊢ ((𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} ∧ 𝐵 = {1, 2}) → dom {⟨2, 6⟩} ⊆ 𝐵) |
17 | relssres 5983 | . . . . 5 ⊢ ((Rel {⟨2, 6⟩} ∧ dom {⟨2, 6⟩} ⊆ 𝐵) → ({⟨2, 6⟩} ↾ 𝐵) = {⟨2, 6⟩}) | |
18 | 11, 16, 17 | sylancr 588 | . . . 4 ⊢ ((𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} ∧ 𝐵 = {1, 2}) → ({⟨2, 6⟩} ↾ 𝐵) = {⟨2, 6⟩}) |
19 | 1re 11162 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
20 | 1lt3 12333 | . . . . . . . 8 ⊢ 1 < 3 | |
21 | 19, 20 | gtneii 11274 | . . . . . . 7 ⊢ 3 ≠ 1 |
22 | 2lt3 12332 | . . . . . . . 8 ⊢ 2 < 3 | |
23 | 7, 22 | gtneii 11274 | . . . . . . 7 ⊢ 3 ≠ 2 |
24 | 21, 23 | nelpri 4620 | . . . . . 6 ⊢ ¬ 3 ∈ {1, 2} |
25 | 14 | eleq2d 2824 | . . . . . 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 7104 | . . . . 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 4129 | . . 3 ⊢ ((𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} ∧ 𝐵 = {1, 2}) → (({⟨2, 6⟩} ↾ 𝐵) ∪ ({⟨3, 9⟩} ↾ 𝐵)) = ({⟨2, 6⟩} ∪ ∅)) |
30 | un0 4355 | . . 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 397 = wceq 1542 ∈ wcel 2107 ∪ cun 3913 ⊆ wss 3915 ∅c0 4287 {csn 4591 {cpr 4593 ⟨cop 4597 dom cdm 5638 ↾ cres 5640 Rel wrel 5643 ℝcr 11057 1c1 11059 2c2 12215 3c3 12216 6c6 12219 9c9 12222 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-po 5550 df-so 5551 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 |
This theorem is referenced by: ex-ima 29428 |
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