Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > fvtp1g | 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 |
---|---|
fvtp1g | ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶)) → ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉}‘𝐴) = 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-tp 3530 | . . 3 ⊢ {〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉} = ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉} ∪ {〈𝐶, 𝐹〉}) | |
2 | 1 | fveq1i 5415 | . 2 ⊢ ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉}‘𝐴) = (({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉} ∪ {〈𝐶, 𝐹〉})‘𝐴) |
3 | necom 2390 | . . . . 5 ⊢ (𝐴 ≠ 𝐶 ↔ 𝐶 ≠ 𝐴) | |
4 | fvunsng 5607 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ≠ 𝐴) → (({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉} ∪ {〈𝐶, 𝐹〉})‘𝐴) = ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉}‘𝐴)) | |
5 | 3, 4 | sylan2b 285 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) → (({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉} ∪ {〈𝐶, 𝐹〉})‘𝐴) = ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉}‘𝐴)) |
6 | 5 | ad2ant2rl 502 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶)) → (({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉} ∪ {〈𝐶, 𝐹〉})‘𝐴) = ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉}‘𝐴)) |
7 | fvpr1g 5619 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉}‘𝐴) = 𝐷) | |
8 | 7 | 3expa 1181 | . . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) ∧ 𝐴 ≠ 𝐵) → ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉}‘𝐴) = 𝐷) |
9 | 8 | adantrr 470 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶)) → ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉}‘𝐴) = 𝐷) |
10 | 6, 9 | eqtrd 2170 | . 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶)) → (({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉} ∪ {〈𝐶, 𝐹〉})‘𝐴) = 𝐷) |
11 | 2, 10 | syl5eq 2182 | 1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶)) → ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉}‘𝐴) = 𝐷) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 ≠ wne 2306 ∪ cun 3064 {csn 3522 {cpr 3523 {ctp 3524 〈cop 3525 ‘cfv 5118 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-tp 3530 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-res 4546 df-iota 5083 df-fun 5120 df-fv 5126 |
This theorem is referenced by: fvtp2g 5622 fvtp1 5624 |
Copyright terms: Public domain | W3C validator |