Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapfzcons1 | Structured version Visualization version GIF version |
Description: Recover prefix mapping from an extended mapping. (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.) |
Ref | Expression |
---|---|
mapfzcons.1 | ⊢ 𝑀 = (𝑁 + 1) |
Ref | Expression |
---|---|
mapfzcons1 | ⊢ (𝐴 ∈ (𝐵 ↑m (1...𝑁)) → ((𝐴 ∪ {〈𝑀, 𝐶〉}) ↾ (1...𝑁)) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elmapi 8418 | . . . 4 ⊢ (𝐴 ∈ (𝐵 ↑m (1...𝑁)) → 𝐴:(1...𝑁)⟶𝐵) | |
2 | ffn 6508 | . . . 4 ⊢ (𝐴:(1...𝑁)⟶𝐵 → 𝐴 Fn (1...𝑁)) | |
3 | fnresdm 6460 | . . . 4 ⊢ (𝐴 Fn (1...𝑁) → (𝐴 ↾ (1...𝑁)) = 𝐴) | |
4 | 1, 2, 3 | 3syl 18 | . . 3 ⊢ (𝐴 ∈ (𝐵 ↑m (1...𝑁)) → (𝐴 ↾ (1...𝑁)) = 𝐴) |
5 | 4 | uneq1d 4137 | . 2 ⊢ (𝐴 ∈ (𝐵 ↑m (1...𝑁)) → ((𝐴 ↾ (1...𝑁)) ∪ ({〈𝑀, 𝐶〉} ↾ (1...𝑁))) = (𝐴 ∪ ({〈𝑀, 𝐶〉} ↾ (1...𝑁)))) |
6 | resundir 5862 | . 2 ⊢ ((𝐴 ∪ {〈𝑀, 𝐶〉}) ↾ (1...𝑁)) = ((𝐴 ↾ (1...𝑁)) ∪ ({〈𝑀, 𝐶〉} ↾ (1...𝑁))) | |
7 | dmres 5869 | . . . . . 6 ⊢ dom ({〈𝑀, 𝐶〉} ↾ (1...𝑁)) = ((1...𝑁) ∩ dom {〈𝑀, 𝐶〉}) | |
8 | dmsnopss 6065 | . . . . . . . . 9 ⊢ dom {〈𝑀, 𝐶〉} ⊆ {𝑀} | |
9 | mapfzcons.1 | . . . . . . . . . 10 ⊢ 𝑀 = (𝑁 + 1) | |
10 | 9 | sneqi 4570 | . . . . . . . . 9 ⊢ {𝑀} = {(𝑁 + 1)} |
11 | 8, 10 | sseqtri 4002 | . . . . . . . 8 ⊢ dom {〈𝑀, 𝐶〉} ⊆ {(𝑁 + 1)} |
12 | sslin 4210 | . . . . . . . 8 ⊢ (dom {〈𝑀, 𝐶〉} ⊆ {(𝑁 + 1)} → ((1...𝑁) ∩ dom {〈𝑀, 𝐶〉}) ⊆ ((1...𝑁) ∩ {(𝑁 + 1)})) | |
13 | 11, 12 | ax-mp 5 | . . . . . . 7 ⊢ ((1...𝑁) ∩ dom {〈𝑀, 𝐶〉}) ⊆ ((1...𝑁) ∩ {(𝑁 + 1)}) |
14 | fzp1disj 12956 | . . . . . . 7 ⊢ ((1...𝑁) ∩ {(𝑁 + 1)}) = ∅ | |
15 | sseq0 4352 | . . . . . . 7 ⊢ ((((1...𝑁) ∩ dom {〈𝑀, 𝐶〉}) ⊆ ((1...𝑁) ∩ {(𝑁 + 1)}) ∧ ((1...𝑁) ∩ {(𝑁 + 1)}) = ∅) → ((1...𝑁) ∩ dom {〈𝑀, 𝐶〉}) = ∅) | |
16 | 13, 14, 15 | mp2an 688 | . . . . . 6 ⊢ ((1...𝑁) ∩ dom {〈𝑀, 𝐶〉}) = ∅ |
17 | 7, 16 | eqtri 2844 | . . . . 5 ⊢ dom ({〈𝑀, 𝐶〉} ↾ (1...𝑁)) = ∅ |
18 | relres 5876 | . . . . . 6 ⊢ Rel ({〈𝑀, 𝐶〉} ↾ (1...𝑁)) | |
19 | reldm0 5792 | . . . . . 6 ⊢ (Rel ({〈𝑀, 𝐶〉} ↾ (1...𝑁)) → (({〈𝑀, 𝐶〉} ↾ (1...𝑁)) = ∅ ↔ dom ({〈𝑀, 𝐶〉} ↾ (1...𝑁)) = ∅)) | |
20 | 18, 19 | ax-mp 5 | . . . . 5 ⊢ (({〈𝑀, 𝐶〉} ↾ (1...𝑁)) = ∅ ↔ dom ({〈𝑀, 𝐶〉} ↾ (1...𝑁)) = ∅) |
21 | 17, 20 | mpbir 232 | . . . 4 ⊢ ({〈𝑀, 𝐶〉} ↾ (1...𝑁)) = ∅ |
22 | 21 | uneq2i 4135 | . . 3 ⊢ (𝐴 ∪ ({〈𝑀, 𝐶〉} ↾ (1...𝑁))) = (𝐴 ∪ ∅) |
23 | un0 4343 | . . 3 ⊢ (𝐴 ∪ ∅) = 𝐴 | |
24 | 22, 23 | eqtr2i 2845 | . 2 ⊢ 𝐴 = (𝐴 ∪ ({〈𝑀, 𝐶〉} ↾ (1...𝑁))) |
25 | 5, 6, 24 | 3eqtr4g 2881 | 1 ⊢ (𝐴 ∈ (𝐵 ↑m (1...𝑁)) → ((𝐴 ∪ {〈𝑀, 𝐶〉}) ↾ (1...𝑁)) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1528 ∈ wcel 2105 ∪ cun 3933 ∩ cin 3934 ⊆ wss 3935 ∅c0 4290 {csn 4559 〈cop 4565 dom cdm 5549 ↾ cres 5551 Rel wrel 5554 Fn wfn 6344 ⟶wf 6345 (class class class)co 7145 ↑m cmap 8396 1c1 10527 + caddc 10529 ...cfz 12882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 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 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-iun 4914 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 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7680 df-2nd 7681 df-er 8279 df-map 8398 df-en 8499 df-dom 8500 df-sdom 8501 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-z 11971 df-uz 12233 df-fz 12883 |
This theorem is referenced by: rexrabdioph 39271 |
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