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Mirrors > Home > MPE Home > Th. List > fvtp3 | Structured version Visualization version GIF version |
Description: The third value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) |
Ref | Expression |
---|---|
fvtp3.1 | ⊢ 𝐶 ∈ V |
fvtp3.4 | ⊢ 𝐹 ∈ V |
Ref | Expression |
---|---|
fvtp3 | ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉}‘𝐶) = 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tprot 4685 | . . 3 ⊢ {〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉} = {〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉, 〈𝐴, 𝐷〉} | |
2 | 1 | fveq1i 6775 | . 2 ⊢ ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉}‘𝐶) = ({〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉, 〈𝐴, 𝐷〉}‘𝐶) |
3 | necom 2997 | . . . 4 ⊢ (𝐴 ≠ 𝐶 ↔ 𝐶 ≠ 𝐴) | |
4 | fvtp3.1 | . . . . 5 ⊢ 𝐶 ∈ V | |
5 | fvtp3.4 | . . . . 5 ⊢ 𝐹 ∈ V | |
6 | 4, 5 | fvtp2 7071 | . . . 4 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴) → ({〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉, 〈𝐴, 𝐷〉}‘𝐶) = 𝐹) |
7 | 3, 6 | sylan2b 594 | . . 3 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐴 ≠ 𝐶) → ({〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉, 〈𝐴, 𝐷〉}‘𝐶) = 𝐹) |
8 | 7 | ancoms 459 | . 2 ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉, 〈𝐴, 𝐷〉}‘𝐶) = 𝐹) |
9 | 2, 8 | eqtrid 2790 | 1 ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉}‘𝐶) = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 Vcvv 3432 {ctp 4565 〈cop 4567 ‘cfv 6433 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-res 5601 df-iota 6391 df-fun 6435 df-fv 6441 |
This theorem is referenced by: fntpb 7085 rabren3dioph 40637 nnsum4primesodd 45248 nnsum4primesoddALTV 45249 |
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