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Mirrors > Home > MPE Home > Th. List > fntp | Structured version Visualization version GIF version |
Description: A function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
fntp.1 | ⊢ 𝐴 ∈ V |
fntp.2 | ⊢ 𝐵 ∈ V |
fntp.3 | ⊢ 𝐶 ∈ V |
fntp.4 | ⊢ 𝐷 ∈ V |
fntp.5 | ⊢ 𝐸 ∈ V |
fntp.6 | ⊢ 𝐹 ∈ V |
Ref | Expression |
---|---|
fntp | ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → {〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉} Fn {𝐴, 𝐵, 𝐶}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fntp.1 | . . 3 ⊢ 𝐴 ∈ V | |
2 | fntp.2 | . . 3 ⊢ 𝐵 ∈ V | |
3 | fntp.3 | . . 3 ⊢ 𝐶 ∈ V | |
4 | fntp.4 | . . 3 ⊢ 𝐷 ∈ V | |
5 | fntp.5 | . . 3 ⊢ 𝐸 ∈ V | |
6 | fntp.6 | . . 3 ⊢ 𝐹 ∈ V | |
7 | 1, 2, 3, 4, 5, 6 | funtp 6242 | . 2 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → Fun {〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉}) |
8 | 4, 5, 6 | dmtpop 5912 | . 2 ⊢ dom {〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉} = {𝐴, 𝐵, 𝐶} |
9 | df-fn 6189 | . 2 ⊢ ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉} Fn {𝐴, 𝐵, 𝐶} ↔ (Fun {〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉} ∧ dom {〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉} = {𝐴, 𝐵, 𝐶})) | |
10 | 7, 8, 9 | sylanblrc 582 | 1 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → {〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉} Fn {𝐴, 𝐵, 𝐶}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1069 = wceq 1508 ∈ wcel 2051 ≠ wne 2962 Vcvv 3410 {ctp 4440 〈cop 4442 dom cdm 5404 Fun wfun 6180 Fn wfn 6181 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-sep 5057 ax-nul 5064 ax-pr 5183 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-ral 3088 df-rex 3089 df-rab 3092 df-v 3412 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-nul 4174 df-if 4346 df-sn 4437 df-pr 4439 df-tp 4441 df-op 4443 df-br 4927 df-opab 4989 df-id 5309 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-fun 6188 df-fn 6189 |
This theorem is referenced by: fntpb 6797 rabren3dioph 38842 |
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