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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 1424 | . 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 1424 | . 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 1424 | . 2 ⊢ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) |
13 | ftpg 6586 | . 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 1573 | 1 ⊢ {〈𝐴, 𝑋〉, 〈𝐵, 𝑌〉, 〈𝐶, 𝑍〉}:{𝐴, 𝐵, 𝐶}⟶{𝑋, 𝑌, 𝑍} |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1072 ∈ wcel 2139 ≠ wne 2932 Vcvv 3340 {ctp 4325 〈cop 4327 ⟶wf 6045 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pr 5055 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-br 4805 df-opab 4865 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 |
This theorem is referenced by: rabren3dioph 37881 nnsum4primesodd 42194 nnsum4primesoddALTV 42195 |
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