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Mirrors > Home > MPE Home > Th. List > fvtp2g | Structured version Visualization version GIF version |
Description: The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.) |
Ref | Expression |
---|---|
fvtp2g | ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶)) → ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉}‘𝐵) = 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tprot 4645 | . . 3 ⊢ {〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉} = {〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉, 〈𝐴, 𝐷〉} | |
2 | 1 | fveq1i 6646 | . 2 ⊢ ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉}‘𝐵) = ({〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉, 〈𝐴, 𝐷〉}‘𝐵) |
3 | necom 3040 | . . . 4 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴) | |
4 | fvtp1g 6937 | . . . . . 6 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐴)) → ({〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉, 〈𝐴, 𝐷〉}‘𝐵) = 𝐸) | |
5 | 4 | expcom 417 | . . . . 5 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐴) → ((𝐵 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → ({〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉, 〈𝐴, 𝐷〉}‘𝐵) = 𝐸)) |
6 | 5 | ancoms 462 | . . . 4 ⊢ ((𝐵 ≠ 𝐴 ∧ 𝐵 ≠ 𝐶) → ((𝐵 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → ({〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉, 〈𝐴, 𝐷〉}‘𝐵) = 𝐸)) |
7 | 3, 6 | sylanb 584 | . . 3 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) → ((𝐵 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → ({〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉, 〈𝐴, 𝐷〉}‘𝐵) = 𝐸)) |
8 | 7 | impcom 411 | . 2 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶)) → ({〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉, 〈𝐴, 𝐷〉}‘𝐵) = 𝐸) |
9 | 2, 8 | syl5eq 2845 | 1 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶)) → ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉}‘𝐵) = 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 {ctp 4529 〈cop 4531 ‘cfv 6324 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-res 5531 df-iota 6283 df-fun 6326 df-fv 6332 |
This theorem is referenced by: fvtp3g 6939 |
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