![]() |
Mathbox for Stefan O'Rear |
< Previous
Next >
Nearby theorems |
|
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 8878 | . . . 4 ⊢ (𝐴 ∈ (𝐵 ↑m (1...𝑁)) → 𝐴:(1...𝑁)⟶𝐵) | |
2 | ffn 6728 | . . . 4 ⊢ (𝐴:(1...𝑁)⟶𝐵 → 𝐴 Fn (1...𝑁)) | |
3 | fnresdm 6680 | . . . 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 4162 | . 2 ⊢ (𝐴 ∈ (𝐵 ↑m (1...𝑁)) → ((𝐴 ↾ (1...𝑁)) ∪ ({〈𝑀, 𝐶〉} ↾ (1...𝑁))) = (𝐴 ∪ ({〈𝑀, 𝐶〉} ↾ (1...𝑁)))) |
6 | resundir 6004 | . 2 ⊢ ((𝐴 ∪ {〈𝑀, 𝐶〉}) ↾ (1...𝑁)) = ((𝐴 ↾ (1...𝑁)) ∪ ({〈𝑀, 𝐶〉} ↾ (1...𝑁))) | |
7 | dmres 6021 | . . . . . 6 ⊢ dom ({〈𝑀, 𝐶〉} ↾ (1...𝑁)) = ((1...𝑁) ∩ dom {〈𝑀, 𝐶〉}) | |
8 | dmsnopss 6225 | . . . . . . . . 9 ⊢ dom {〈𝑀, 𝐶〉} ⊆ {𝑀} | |
9 | mapfzcons.1 | . . . . . . . . . 10 ⊢ 𝑀 = (𝑁 + 1) | |
10 | 9 | sneqi 4644 | . . . . . . . . 9 ⊢ {𝑀} = {(𝑁 + 1)} |
11 | 8, 10 | sseqtri 4016 | . . . . . . . 8 ⊢ dom {〈𝑀, 𝐶〉} ⊆ {(𝑁 + 1)} |
12 | sslin 4236 | . . . . . . . 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 13614 | . . . . . . 7 ⊢ ((1...𝑁) ∩ {(𝑁 + 1)}) = ∅ | |
15 | sseq0 4404 | . . . . . . 7 ⊢ ((((1...𝑁) ∩ dom {〈𝑀, 𝐶〉}) ⊆ ((1...𝑁) ∩ {(𝑁 + 1)}) ∧ ((1...𝑁) ∩ {(𝑁 + 1)}) = ∅) → ((1...𝑁) ∩ dom {〈𝑀, 𝐶〉}) = ∅) | |
16 | 13, 14, 15 | mp2an 690 | . . . . . 6 ⊢ ((1...𝑁) ∩ dom {〈𝑀, 𝐶〉}) = ∅ |
17 | 7, 16 | eqtri 2754 | . . . . 5 ⊢ dom ({〈𝑀, 𝐶〉} ↾ (1...𝑁)) = ∅ |
18 | relres 6015 | . . . . . 6 ⊢ Rel ({〈𝑀, 𝐶〉} ↾ (1...𝑁)) | |
19 | reldm0 5934 | . . . . . 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 230 | . . . 4 ⊢ ({〈𝑀, 𝐶〉} ↾ (1...𝑁)) = ∅ |
22 | 21 | uneq2i 4160 | . . 3 ⊢ (𝐴 ∪ ({〈𝑀, 𝐶〉} ↾ (1...𝑁))) = (𝐴 ∪ ∅) |
23 | un0 4395 | . . 3 ⊢ (𝐴 ∪ ∅) = 𝐴 | |
24 | 22, 23 | eqtr2i 2755 | . 2 ⊢ 𝐴 = (𝐴 ∪ ({〈𝑀, 𝐶〉} ↾ (1...𝑁))) |
25 | 5, 6, 24 | 3eqtr4g 2791 | 1 ⊢ (𝐴 ∈ (𝐵 ↑m (1...𝑁)) → ((𝐴 ∪ {〈𝑀, 𝐶〉}) ↾ (1...𝑁)) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1534 ∈ wcel 2099 ∪ cun 3945 ∩ cin 3946 ⊆ wss 3947 ∅c0 4325 {csn 4633 〈cop 4639 dom cdm 5682 ↾ cres 5684 Rel wrel 5687 Fn wfn 6549 ⟶wf 6550 (class class class)co 7424 ↑m cmap 8855 1c1 11159 + caddc 11161 ...cfz 13538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 8003 df-2nd 8004 df-er 8734 df-map 8857 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-z 12611 df-uz 12875 df-fz 13539 |
This theorem is referenced by: rexrabdioph 42451 |
Copyright terms: Public domain | W3C validator |