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Mirrors > Home > MPE Home > Th. List > fvtp2 | Structured version Visualization version GIF version |
Description: The second value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) |
Ref | Expression |
---|---|
fvtp2.1 | ⊢ 𝐵 ∈ V |
fvtp2.4 | ⊢ 𝐸 ∈ V |
Ref | Expression |
---|---|
fvtp2 | ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) → ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉}‘𝐵) = 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tprot 4678 | . . 3 ⊢ {〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉} = {〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉, 〈𝐴, 𝐷〉} | |
2 | 1 | fveq1i 6665 | . 2 ⊢ ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉}‘𝐵) = ({〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉, 〈𝐴, 𝐷〉}‘𝐵) |
3 | necom 3069 | . . 3 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴) | |
4 | fvtp2.1 | . . . . 5 ⊢ 𝐵 ∈ V | |
5 | fvtp2.4 | . . . . 5 ⊢ 𝐸 ∈ V | |
6 | 4, 5 | fvtp1 6951 | . . . 4 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐴) → ({〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉, 〈𝐴, 𝐷〉}‘𝐵) = 𝐸) |
7 | 6 | ancoms 461 | . . 3 ⊢ ((𝐵 ≠ 𝐴 ∧ 𝐵 ≠ 𝐶) → ({〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉, 〈𝐴, 𝐷〉}‘𝐵) = 𝐸) |
8 | 3, 7 | sylanb 583 | . 2 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) → ({〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉, 〈𝐴, 𝐷〉}‘𝐵) = 𝐸) |
9 | 2, 8 | syl5eq 2868 | 1 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) → ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉}‘𝐵) = 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 Vcvv 3494 {ctp 4564 〈cop 4566 ‘cfv 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 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 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-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-res 5561 df-iota 6308 df-fun 6351 df-fv 6357 |
This theorem is referenced by: fvtp3 6953 fntpb 6966 rabren3dioph 39405 nnsum4primesodd 43955 nnsum4primesoddALTV 43956 |
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