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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapfzcons2 | Structured version Visualization version GIF version |
Description: Recover added element 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 |
---|---|
mapfzcons2 | ⊢ ((𝐴 ∈ (𝐵 ↑𝑚 (1...𝑁)) ∧ 𝐶 ∈ 𝐵) → ((𝐴 ∪ {〈𝑀, 𝐶〉})‘𝑀) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapfzcons.1 | . . . 4 ⊢ 𝑀 = (𝑁 + 1) | |
2 | ovex 6936 | . . . 4 ⊢ (𝑁 + 1) ∈ V | |
3 | 1, 2 | eqeltri 2901 | . . 3 ⊢ 𝑀 ∈ V |
4 | 3 | a1i 11 | . 2 ⊢ ((𝐴 ∈ (𝐵 ↑𝑚 (1...𝑁)) ∧ 𝐶 ∈ 𝐵) → 𝑀 ∈ V) |
5 | elex 3428 | . . 3 ⊢ (𝐶 ∈ 𝐵 → 𝐶 ∈ V) | |
6 | 5 | adantl 475 | . 2 ⊢ ((𝐴 ∈ (𝐵 ↑𝑚 (1...𝑁)) ∧ 𝐶 ∈ 𝐵) → 𝐶 ∈ V) |
7 | elmapi 8143 | . . . . . . 7 ⊢ (𝐴 ∈ (𝐵 ↑𝑚 (1...𝑁)) → 𝐴:(1...𝑁)⟶𝐵) | |
8 | 7 | fdmd 6286 | . . . . . 6 ⊢ (𝐴 ∈ (𝐵 ↑𝑚 (1...𝑁)) → dom 𝐴 = (1...𝑁)) |
9 | 8 | adantr 474 | . . . . 5 ⊢ ((𝐴 ∈ (𝐵 ↑𝑚 (1...𝑁)) ∧ 𝐶 ∈ 𝐵) → dom 𝐴 = (1...𝑁)) |
10 | 9 | ineq1d 4039 | . . . 4 ⊢ ((𝐴 ∈ (𝐵 ↑𝑚 (1...𝑁)) ∧ 𝐶 ∈ 𝐵) → (dom 𝐴 ∩ {𝑀}) = ((1...𝑁) ∩ {𝑀})) |
11 | 1 | sneqi 4407 | . . . . . 6 ⊢ {𝑀} = {(𝑁 + 1)} |
12 | 11 | ineq2i 4037 | . . . . 5 ⊢ ((1...𝑁) ∩ {𝑀}) = ((1...𝑁) ∩ {(𝑁 + 1)}) |
13 | fzp1disj 12692 | . . . . 5 ⊢ ((1...𝑁) ∩ {(𝑁 + 1)}) = ∅ | |
14 | 12, 13 | eqtri 2848 | . . . 4 ⊢ ((1...𝑁) ∩ {𝑀}) = ∅ |
15 | 10, 14 | syl6eq 2876 | . . 3 ⊢ ((𝐴 ∈ (𝐵 ↑𝑚 (1...𝑁)) ∧ 𝐶 ∈ 𝐵) → (dom 𝐴 ∩ {𝑀}) = ∅) |
16 | disjsn 4464 | . . 3 ⊢ ((dom 𝐴 ∩ {𝑀}) = ∅ ↔ ¬ 𝑀 ∈ dom 𝐴) | |
17 | 15, 16 | sylib 210 | . 2 ⊢ ((𝐴 ∈ (𝐵 ↑𝑚 (1...𝑁)) ∧ 𝐶 ∈ 𝐵) → ¬ 𝑀 ∈ dom 𝐴) |
18 | fsnunfv 6708 | . 2 ⊢ ((𝑀 ∈ V ∧ 𝐶 ∈ V ∧ ¬ 𝑀 ∈ dom 𝐴) → ((𝐴 ∪ {〈𝑀, 𝐶〉})‘𝑀) = 𝐶) | |
19 | 4, 6, 17, 18 | syl3anc 1496 | 1 ⊢ ((𝐴 ∈ (𝐵 ↑𝑚 (1...𝑁)) ∧ 𝐶 ∈ 𝐵) → ((𝐴 ∪ {〈𝑀, 𝐶〉})‘𝑀) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 Vcvv 3413 ∪ cun 3795 ∩ cin 3796 ∅c0 4143 {csn 4396 〈cop 4402 dom cdm 5341 ‘cfv 6122 (class class class)co 6904 ↑𝑚 cmap 8121 1c1 10252 + caddc 10254 ...cfz 12618 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-cnex 10307 ax-resscn 10308 ax-1cn 10309 ax-icn 10310 ax-addcl 10311 ax-addrcl 10312 ax-mulcl 10313 ax-mulrcl 10314 ax-mulcom 10315 ax-addass 10316 ax-mulass 10317 ax-distr 10318 ax-i2m1 10319 ax-1ne0 10320 ax-1rid 10321 ax-rnegex 10322 ax-rrecex 10323 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326 ax-pre-ltadd 10327 ax-pre-mulgt0 10328 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-reu 3123 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-op 4403 df-uni 4658 df-iun 4741 df-br 4873 df-opab 4935 df-mpt 4952 df-id 5249 df-po 5262 df-so 5263 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-riota 6865 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-1st 7427 df-2nd 7428 df-er 8008 df-map 8123 df-en 8222 df-dom 8223 df-sdom 8224 df-pnf 10392 df-mnf 10393 df-xr 10394 df-ltxr 10395 df-le 10396 df-sub 10586 df-neg 10587 df-z 11704 df-uz 11968 df-fz 12619 |
This theorem is referenced by: rexrabdioph 38201 |
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