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Mirrors > Home > MPE Home > Th. List > ftp | Structured version Visualization version GIF version |
Description: A function with a domain of three elements. (Contributed by Stefan O'Rear, 17-Oct-2014.) (Proof shortened by Alexander van der Vekens, 23-Jan-2018.) |
Ref | Expression |
---|---|
ftp.a | ⊢ 𝐴 ∈ V |
ftp.b | ⊢ 𝐵 ∈ V |
ftp.c | ⊢ 𝐶 ∈ V |
ftp.d | ⊢ 𝑋 ∈ V |
ftp.e | ⊢ 𝑌 ∈ V |
ftp.f | ⊢ 𝑍 ∈ V |
ftp.g | ⊢ 𝐴 ≠ 𝐵 |
ftp.h | ⊢ 𝐴 ≠ 𝐶 |
ftp.i | ⊢ 𝐵 ≠ 𝐶 |
Ref | Expression |
---|---|
ftp | ⊢ {〈𝐴, 𝑋〉, 〈𝐵, 𝑌〉, 〈𝐶, 𝑍〉}:{𝐴, 𝐵, 𝐶}⟶{𝑋, 𝑌, 𝑍} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ftp.a | . . 3 ⊢ 𝐴 ∈ V | |
2 | ftp.b | . . 3 ⊢ 𝐵 ∈ V | |
3 | ftp.c | . . 3 ⊢ 𝐶 ∈ V | |
4 | 1, 2, 3 | 3pm3.2i 1340 | . 2 ⊢ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) |
5 | ftp.d | . . 3 ⊢ 𝑋 ∈ V | |
6 | ftp.e | . . 3 ⊢ 𝑌 ∈ V | |
7 | ftp.f | . . 3 ⊢ 𝑍 ∈ V | |
8 | 5, 6, 7 | 3pm3.2i 1340 | . 2 ⊢ (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ V) |
9 | ftp.g | . . 3 ⊢ 𝐴 ≠ 𝐵 | |
10 | ftp.h | . . 3 ⊢ 𝐴 ≠ 𝐶 | |
11 | ftp.i | . . 3 ⊢ 𝐵 ≠ 𝐶 | |
12 | 9, 10, 11 | 3pm3.2i 1340 | . 2 ⊢ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) |
13 | ftpg 6922 | . 2 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) ∧ (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ V) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → {〈𝐴, 𝑋〉, 〈𝐵, 𝑌〉, 〈𝐶, 𝑍〉}:{𝐴, 𝐵, 𝐶}⟶{𝑋, 𝑌, 𝑍}) | |
14 | 4, 8, 12, 13 | mp3an 1462 | 1 ⊢ {〈𝐴, 𝑋〉, 〈𝐵, 𝑌〉, 〈𝐶, 𝑍〉}:{𝐴, 𝐵, 𝐶}⟶{𝑋, 𝑌, 𝑍} |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1088 ∈ wcel 2113 ≠ wne 2934 Vcvv 3397 {ctp 4517 〈cop 4519 ⟶wf 6329 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pr 5293 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3399 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-br 5028 df-opab 5090 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 |
This theorem is referenced by: rabren3dioph 40193 nnsum4primesodd 44766 nnsum4primesoddALTV 44767 |
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