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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 5859 | . 2 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → Fun {〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉}) |
8 | 4, 5, 6 | dmtpop 5529 | . . 3 ⊢ dom {〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉} = {𝐴, 𝐵, 𝐶} |
9 | 8 | a1i 11 | . 2 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → dom {〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉} = {𝐴, 𝐵, 𝐶}) |
10 | df-fn 5807 | . 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})) | |
11 | 7, 9, 10 | sylanbrc 695 | 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 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 {ctp 4129 〈cop 4131 dom cdm 5038 Fun wfun 5798 Fn wfn 5799 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-fun 5806 df-fn 5807 |
This theorem is referenced by: fntpb 6378 rabren3dioph 36397 |
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