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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 8411 | . . . 4 ⊢ (𝐴 ∈ (𝐵 ↑m (1...𝑁)) → 𝐴:(1...𝑁)⟶𝐵) | |
2 | ffn 6487 | . . . 4 ⊢ (𝐴:(1...𝑁)⟶𝐵 → 𝐴 Fn (1...𝑁)) | |
3 | fnresdm 6438 | . . . 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 4089 | . 2 ⊢ (𝐴 ∈ (𝐵 ↑m (1...𝑁)) → ((𝐴 ↾ (1...𝑁)) ∪ ({〈𝑀, 𝐶〉} ↾ (1...𝑁))) = (𝐴 ∪ ({〈𝑀, 𝐶〉} ↾ (1...𝑁)))) |
6 | resundir 5833 | . 2 ⊢ ((𝐴 ∪ {〈𝑀, 𝐶〉}) ↾ (1...𝑁)) = ((𝐴 ↾ (1...𝑁)) ∪ ({〈𝑀, 𝐶〉} ↾ (1...𝑁))) | |
7 | dmres 5840 | . . . . . 6 ⊢ dom ({〈𝑀, 𝐶〉} ↾ (1...𝑁)) = ((1...𝑁) ∩ dom {〈𝑀, 𝐶〉}) | |
8 | dmsnopss 6038 | . . . . . . . . 9 ⊢ dom {〈𝑀, 𝐶〉} ⊆ {𝑀} | |
9 | mapfzcons.1 | . . . . . . . . . 10 ⊢ 𝑀 = (𝑁 + 1) | |
10 | 9 | sneqi 4536 | . . . . . . . . 9 ⊢ {𝑀} = {(𝑁 + 1)} |
11 | 8, 10 | sseqtri 3951 | . . . . . . . 8 ⊢ dom {〈𝑀, 𝐶〉} ⊆ {(𝑁 + 1)} |
12 | sslin 4161 | . . . . . . . 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 12961 | . . . . . . 7 ⊢ ((1...𝑁) ∩ {(𝑁 + 1)}) = ∅ | |
15 | sseq0 4307 | . . . . . . 7 ⊢ ((((1...𝑁) ∩ dom {〈𝑀, 𝐶〉}) ⊆ ((1...𝑁) ∩ {(𝑁 + 1)}) ∧ ((1...𝑁) ∩ {(𝑁 + 1)}) = ∅) → ((1...𝑁) ∩ dom {〈𝑀, 𝐶〉}) = ∅) | |
16 | 13, 14, 15 | mp2an 691 | . . . . . 6 ⊢ ((1...𝑁) ∩ dom {〈𝑀, 𝐶〉}) = ∅ |
17 | 7, 16 | eqtri 2821 | . . . . 5 ⊢ dom ({〈𝑀, 𝐶〉} ↾ (1...𝑁)) = ∅ |
18 | relres 5847 | . . . . . 6 ⊢ Rel ({〈𝑀, 𝐶〉} ↾ (1...𝑁)) | |
19 | reldm0 5762 | . . . . . 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 234 | . . . 4 ⊢ ({〈𝑀, 𝐶〉} ↾ (1...𝑁)) = ∅ |
22 | 21 | uneq2i 4087 | . . 3 ⊢ (𝐴 ∪ ({〈𝑀, 𝐶〉} ↾ (1...𝑁))) = (𝐴 ∪ ∅) |
23 | un0 4298 | . . 3 ⊢ (𝐴 ∪ ∅) = 𝐴 | |
24 | 22, 23 | eqtr2i 2822 | . 2 ⊢ 𝐴 = (𝐴 ∪ ({〈𝑀, 𝐶〉} ↾ (1...𝑁))) |
25 | 5, 6, 24 | 3eqtr4g 2858 | 1 ⊢ (𝐴 ∈ (𝐵 ↑m (1...𝑁)) → ((𝐴 ∪ {〈𝑀, 𝐶〉}) ↾ (1...𝑁)) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 ∪ cun 3879 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 {csn 4525 〈cop 4531 dom cdm 5519 ↾ cres 5521 Rel wrel 5524 Fn wfn 6319 ⟶wf 6320 (class class class)co 7135 ↑m cmap 8389 1c1 10527 + caddc 10529 ...cfz 12885 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 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 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-z 11970 df-uz 12232 df-fz 12886 |
This theorem is referenced by: rexrabdioph 39735 |
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