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| 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 1339 | . 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 1339 | . 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 1339 | . 2 ⊢ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) | 
| 13 | ftpg 7175 | . 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 1086 ∈ wcel 2107 ≠ wne 2939 Vcvv 3479 {ctp 4629 〈cop 4631 ⟶wf 6556 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-br 5143 df-opab 5205 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 | 
| This theorem is referenced by: rabren3dioph 42831 nnsum4primesodd 47788 nnsum4primesoddALTV 47789 | 
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