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 6577 | . . . 4 ⊢ (𝑁 + 1) ∈ V | |
3 | 1, 2 | eqeltri 2684 | . . 3 ⊢ 𝑀 ∈ V |
4 | 3 | a1i 11 | . 2 ⊢ ((𝐴 ∈ (𝐵 ↑𝑚 (1...𝑁)) ∧ 𝐶 ∈ 𝐵) → 𝑀 ∈ V) |
5 | elex 3185 | . . 3 ⊢ (𝐶 ∈ 𝐵 → 𝐶 ∈ V) | |
6 | 5 | adantl 481 | . 2 ⊢ ((𝐴 ∈ (𝐵 ↑𝑚 (1...𝑁)) ∧ 𝐶 ∈ 𝐵) → 𝐶 ∈ V) |
7 | elmapi 7765 | . . . . . . 7 ⊢ (𝐴 ∈ (𝐵 ↑𝑚 (1...𝑁)) → 𝐴:(1...𝑁)⟶𝐵) | |
8 | fdm 5964 | . . . . . . 7 ⊢ (𝐴:(1...𝑁)⟶𝐵 → dom 𝐴 = (1...𝑁)) | |
9 | 7, 8 | syl 17 | . . . . . 6 ⊢ (𝐴 ∈ (𝐵 ↑𝑚 (1...𝑁)) → dom 𝐴 = (1...𝑁)) |
10 | 9 | adantr 480 | . . . . 5 ⊢ ((𝐴 ∈ (𝐵 ↑𝑚 (1...𝑁)) ∧ 𝐶 ∈ 𝐵) → dom 𝐴 = (1...𝑁)) |
11 | 10 | ineq1d 3775 | . . . 4 ⊢ ((𝐴 ∈ (𝐵 ↑𝑚 (1...𝑁)) ∧ 𝐶 ∈ 𝐵) → (dom 𝐴 ∩ {𝑀}) = ((1...𝑁) ∩ {𝑀})) |
12 | 1 | sneqi 4136 | . . . . . 6 ⊢ {𝑀} = {(𝑁 + 1)} |
13 | 12 | ineq2i 3773 | . . . . 5 ⊢ ((1...𝑁) ∩ {𝑀}) = ((1...𝑁) ∩ {(𝑁 + 1)}) |
14 | fzp1disj 12269 | . . . . 5 ⊢ ((1...𝑁) ∩ {(𝑁 + 1)}) = ∅ | |
15 | 13, 14 | eqtri 2632 | . . . 4 ⊢ ((1...𝑁) ∩ {𝑀}) = ∅ |
16 | 11, 15 | syl6eq 2660 | . . 3 ⊢ ((𝐴 ∈ (𝐵 ↑𝑚 (1...𝑁)) ∧ 𝐶 ∈ 𝐵) → (dom 𝐴 ∩ {𝑀}) = ∅) |
17 | disjsn 4192 | . . 3 ⊢ ((dom 𝐴 ∩ {𝑀}) = ∅ ↔ ¬ 𝑀 ∈ dom 𝐴) | |
18 | 16, 17 | sylib 207 | . 2 ⊢ ((𝐴 ∈ (𝐵 ↑𝑚 (1...𝑁)) ∧ 𝐶 ∈ 𝐵) → ¬ 𝑀 ∈ dom 𝐴) |
19 | fsnunfv 6358 | . 2 ⊢ ((𝑀 ∈ V ∧ 𝐶 ∈ V ∧ ¬ 𝑀 ∈ dom 𝐴) → ((𝐴 ∪ {〈𝑀, 𝐶〉})‘𝑀) = 𝐶) | |
20 | 4, 6, 18, 19 | syl3anc 1318 | 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 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∪ cun 3538 ∩ cin 3539 ∅c0 3874 {csn 4125 〈cop 4131 dom cdm 5038 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 1c1 9816 + caddc 9818 ...cfz 12197 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-z 11255 df-uz 11564 df-fz 12198 |
This theorem is referenced by: rexrabdioph 36376 |
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