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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 6602 | . 2 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → Fun {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}) |
8 | 4, 5, 6 | dmtpop 6214 | . 2 ⊢ dom {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩} = {𝐴, 𝐵, 𝐶} |
9 | df-fn 6543 | . 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 590 | 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 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 Vcvv 3474 {ctp 4631 ⟨cop 4633 dom cdm 5675 Fun wfun 6534 Fn wfn 6535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-fun 6542 df-fn 6543 |
This theorem is referenced by: fntpb 7207 rabren3dioph 41538 |
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