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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 1337 | . 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 1337 | . 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 1337 | . 2 ⊢ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) |
13 | ftpg 7159 | . 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 1458 | 1 ⊢ {⟨𝐴, 𝑋⟩, ⟨𝐵, 𝑌⟩, ⟨𝐶, 𝑍⟩}:{𝐴, 𝐵, 𝐶}⟶{𝑋, 𝑌, 𝑍} |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1085 ∈ wcel 2099 ≠ wne 2935 Vcvv 3469 {ctp 4628 ⟨cop 4630 ⟶wf 6538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-ne 2936 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-br 5143 df-opab 5205 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 |
This theorem is referenced by: rabren3dioph 42207 nnsum4primesodd 47108 nnsum4primesoddALTV 47109 |
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