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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 6559 | . 2 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → Fun {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}) |
8 | 4, 5, 6 | dmtpop 6171 | . 2 ⊢ dom {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩} = {𝐴, 𝐵, 𝐶} |
9 | df-fn 6500 | . 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 591 | 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 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 Vcvv 3444 {ctp 4591 ⟨cop 4593 dom cdm 5634 Fun wfun 6491 Fn wfn 6492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-fun 6499 df-fn 6500 |
This theorem is referenced by: fntpb 7160 rabren3dioph 41181 |
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