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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 6623 | . 2 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → Fun {〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉}) |
| 8 | 4, 5, 6 | dmtpop 6238 | . 2 ⊢ dom {〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉} = {𝐴, 𝐵, 𝐶} |
| 9 | df-fn 6564 | . 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 1540 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 {ctp 4630 〈cop 4632 dom cdm 5685 Fun wfun 6555 Fn wfn 6556 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-fun 6563 df-fn 6564 |
| This theorem is referenced by: fntpb 7229 rabren3dioph 42826 |
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