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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 1239 | . 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 1239 | . 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 1239 | . 2 ⊢ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) |
13 | ftpg 6423 | . 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 1424 | 1 ⊢ {〈𝐴, 𝑋〉, 〈𝐵, 𝑌〉, 〈𝐶, 𝑍〉}:{𝐴, 𝐵, 𝐶}⟶{𝑋, 𝑌, 𝑍} |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1037 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 {ctp 4181 〈cop 4183 ⟶wf 5884 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 |
This theorem is referenced by: rabren3dioph 37379 nnsum4primesodd 41684 nnsum4primesoddALTV 41685 |
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