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Mirrors > Home > MPE Home > Th. List > 2trllemG | Structured version Visualization version GIF version |
Description: Lemma 7 for constr2trl 26129. (Contributed by Alexander van der Vekens, 1-Feb-2018.) |
Ref | Expression |
---|---|
2trlX.p | ⊢ 𝑃 = {〈0, 𝐴〉, 〈1, 𝐵〉, 〈2, 𝐶〉} |
Ref | Expression |
---|---|
2trllemG | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 𝑃:(0...2)⟶𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0z 11265 | . . . 4 ⊢ 0 ∈ ℤ | |
2 | 1z 11284 | . . . 4 ⊢ 1 ∈ ℤ | |
3 | 2z 11286 | . . . 4 ⊢ 2 ∈ ℤ | |
4 | 1, 2, 3 | 3pm3.2i 1232 | . . 3 ⊢ (0 ∈ ℤ ∧ 1 ∈ ℤ ∧ 2 ∈ ℤ) |
5 | 0ne1 10965 | . . . 4 ⊢ 0 ≠ 1 | |
6 | 0ne2 11116 | . . . 4 ⊢ 0 ≠ 2 | |
7 | 1ne2 11117 | . . . 4 ⊢ 1 ≠ 2 | |
8 | 5, 6, 7 | 3pm3.2i 1232 | . . 3 ⊢ (0 ≠ 1 ∧ 0 ≠ 2 ∧ 1 ≠ 2) |
9 | ftpg 6328 | . . . 4 ⊢ (((0 ∈ ℤ ∧ 1 ∈ ℤ ∧ 2 ∈ ℤ) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (0 ≠ 1 ∧ 0 ≠ 2 ∧ 1 ≠ 2)) → {〈0, 𝐴〉, 〈1, 𝐵〉, 〈2, 𝐶〉}:{0, 1, 2}⟶{𝐴, 𝐵, 𝐶}) | |
10 | 2trlX.p | . . . . 5 ⊢ 𝑃 = {〈0, 𝐴〉, 〈1, 𝐵〉, 〈2, 𝐶〉} | |
11 | 10 | feq1i 5949 | . . . 4 ⊢ (𝑃:{0, 1, 2}⟶{𝐴, 𝐵, 𝐶} ↔ {〈0, 𝐴〉, 〈1, 𝐵〉, 〈2, 𝐶〉}:{0, 1, 2}⟶{𝐴, 𝐵, 𝐶}) |
12 | 9, 11 | sylibr 223 | . . 3 ⊢ (((0 ∈ ℤ ∧ 1 ∈ ℤ ∧ 2 ∈ ℤ) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (0 ≠ 1 ∧ 0 ≠ 2 ∧ 1 ≠ 2)) → 𝑃:{0, 1, 2}⟶{𝐴, 𝐵, 𝐶}) |
13 | 4, 8, 12 | mp3an13 1407 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 𝑃:{0, 1, 2}⟶{𝐴, 𝐵, 𝐶}) |
14 | tpssi 4309 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → {𝐴, 𝐵, 𝐶} ⊆ 𝑉) | |
15 | fss 5969 | . . 3 ⊢ ((𝑃:{0, 1, 2}⟶{𝐴, 𝐵, 𝐶} ∧ {𝐴, 𝐵, 𝐶} ⊆ 𝑉) → 𝑃:{0, 1, 2}⟶𝑉) | |
16 | fz0tp 12309 | . . . . 5 ⊢ (0...2) = {0, 1, 2} | |
17 | 16 | a1i 11 | . . . 4 ⊢ ((𝑃:{0, 1, 2}⟶{𝐴, 𝐵, 𝐶} ∧ {𝐴, 𝐵, 𝐶} ⊆ 𝑉) → (0...2) = {0, 1, 2}) |
18 | 17 | feq2d 5944 | . . 3 ⊢ ((𝑃:{0, 1, 2}⟶{𝐴, 𝐵, 𝐶} ∧ {𝐴, 𝐵, 𝐶} ⊆ 𝑉) → (𝑃:(0...2)⟶𝑉 ↔ 𝑃:{0, 1, 2}⟶𝑉)) |
19 | 15, 18 | mpbird 246 | . 2 ⊢ ((𝑃:{0, 1, 2}⟶{𝐴, 𝐵, 𝐶} ∧ {𝐴, 𝐵, 𝐶} ⊆ 𝑉) → 𝑃:(0...2)⟶𝑉) |
20 | 13, 14, 19 | syl2anc 691 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 𝑃:(0...2)⟶𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ⊆ wss 3540 {ctp 4129 〈cop 4131 ⟶wf 5800 (class class class)co 6549 0cc0 9815 1c1 9816 2c2 10947 ℤcz 11254 ...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-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 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-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 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-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 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-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 |
This theorem is referenced by: wlkntrl 26092 constr2wlk 26128 constr2trl 26129 |
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