Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 4130 | . . 3 ⊢ {〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉} = ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉} ∪ {〈𝐶, 𝐹〉}) | |
2 | 1 | fveq1i 6104 | . 2 ⊢ ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉}‘𝐴) = (({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉} ∪ {〈𝐶, 𝐹〉})‘𝐴) |
3 | necom 2835 | . . . 4 ⊢ (𝐴 ≠ 𝐶 ↔ 𝐶 ≠ 𝐴) | |
4 | fvunsn 6350 | . . . 4 ⊢ (𝐶 ≠ 𝐴 → (({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉} ∪ {〈𝐶, 𝐹〉})‘𝐴) = ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉}‘𝐴)) | |
5 | 3, 4 | sylbi 206 | . . 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 6361 | . . 3 ⊢ (𝐴 ≠ 𝐵 → ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉}‘𝐴) = 𝐷) |
9 | 5, 8 | sylan9eqr 2666 | . 2 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) → (({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉} ∪ {〈𝐶, 𝐹〉})‘𝐴) = 𝐷) |
10 | 2, 9 | syl5eq 2656 | 1 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) → ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉}‘𝐴) = 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 ∪ cun 3538 {csn 4125 {cpr 4127 {ctp 4129 〈cop 4131 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-res 5050 df-iota 5768 df-fun 5806 df-fv 5812 |
This theorem is referenced by: fvtp2 6366 fntpb 6378 wlkntrllem2 26090 constr3lem5 26176 rabren3dioph 36397 nnsum4primesodd 40212 nnsum4primesoddALTV 40213 |
Copyright terms: Public domain | W3C validator |