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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 6410 | . 2 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → Fun {〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉}) |
8 | 4, 5, 6 | dmtpop 6074 | . 2 ⊢ dom {〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉} = {𝐴, 𝐵, 𝐶} |
9 | df-fn 6357 | . 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 592 | 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 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 Vcvv 3494 {ctp 4570 〈cop 4572 dom cdm 5554 Fun wfun 6348 Fn wfn 6349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-br 5066 df-opab 5128 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-fun 6356 df-fn 6357 |
This theorem is referenced by: fntpb 6971 rabren3dioph 39410 |
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