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Mirrors > Home > MPE Home > Th. List > fvtp1 | Structured version Visualization version GIF version |
Description: The first value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) |
Ref | Expression |
---|---|
fvtp1.1 | ⊢ 𝐴 ∈ V |
fvtp1.4 | ⊢ 𝐷 ∈ V |
Ref | Expression |
---|---|
fvtp1 | ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) → ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉}‘𝐴) = 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-tp 4635 | . . 3 ⊢ {〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉} = ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉} ∪ {〈𝐶, 𝐹〉}) | |
2 | 1 | fveq1i 6907 | . 2 ⊢ ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉}‘𝐴) = (({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉} ∪ {〈𝐶, 𝐹〉})‘𝐴) |
3 | necom 2991 | . . . 4 ⊢ (𝐴 ≠ 𝐶 ↔ 𝐶 ≠ 𝐴) | |
4 | fvunsn 7198 | . . . 4 ⊢ (𝐶 ≠ 𝐴 → (({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉} ∪ {〈𝐶, 𝐹〉})‘𝐴) = ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉}‘𝐴)) | |
5 | 3, 4 | sylbi 217 | . . 3 ⊢ (𝐴 ≠ 𝐶 → (({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉} ∪ {〈𝐶, 𝐹〉})‘𝐴) = ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉}‘𝐴)) |
6 | fvtp1.1 | . . . 4 ⊢ 𝐴 ∈ V | |
7 | fvtp1.4 | . . . 4 ⊢ 𝐷 ∈ V | |
8 | 6, 7 | fvpr1 7211 | . . 3 ⊢ (𝐴 ≠ 𝐵 → ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉}‘𝐴) = 𝐷) |
9 | 5, 8 | sylan9eqr 2796 | . 2 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) → (({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉} ∪ {〈𝐶, 𝐹〉})‘𝐴) = 𝐷) |
10 | 2, 9 | eqtrid 2786 | 1 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) → ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉}‘𝐴) = 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 Vcvv 3477 ∪ cun 3960 {csn 4630 {cpr 4632 {ctp 4634 〈cop 4636 ‘cfv 6562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-res 5700 df-iota 6515 df-fun 6564 df-fv 6570 |
This theorem is referenced by: fvtp2 7215 fntpb 7228 rabren3dioph 42802 nnsum4primesodd 47720 nnsum4primesoddALTV 47721 |
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