Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > fvtp3 | Structured version Visualization version GIF version |
Description: The third value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) |
Ref | Expression |
---|---|
fvtp3.1 | ⊢ 𝐶 ∈ V |
fvtp3.4 | ⊢ 𝐹 ∈ V |
Ref | Expression |
---|---|
fvtp3 | ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉}‘𝐶) = 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tprot 4641 | . . 3 ⊢ {〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉} = {〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉, 〈𝐴, 𝐷〉} | |
2 | 1 | fveq1i 6676 | . 2 ⊢ ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉}‘𝐶) = ({〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉, 〈𝐴, 𝐷〉}‘𝐶) |
3 | necom 2987 | . . . 4 ⊢ (𝐴 ≠ 𝐶 ↔ 𝐶 ≠ 𝐴) | |
4 | fvtp3.1 | . . . . 5 ⊢ 𝐶 ∈ V | |
5 | fvtp3.4 | . . . . 5 ⊢ 𝐹 ∈ V | |
6 | 4, 5 | fvtp2 6969 | . . . 4 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴) → ({〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉, 〈𝐴, 𝐷〉}‘𝐶) = 𝐹) |
7 | 3, 6 | sylan2b 597 | . . 3 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐴 ≠ 𝐶) → ({〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉, 〈𝐴, 𝐷〉}‘𝐶) = 𝐹) |
8 | 7 | ancoms 462 | . 2 ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉, 〈𝐴, 𝐷〉}‘𝐶) = 𝐹) |
9 | 2, 8 | syl5eq 2785 | 1 ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({〈𝐴, 𝐷〉, 〈𝐵, 𝐸〉, 〈𝐶, 𝐹〉}‘𝐶) = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2113 ≠ wne 2934 Vcvv 3398 {ctp 4521 〈cop 4523 ‘cfv 6340 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5168 ax-nul 5175 ax-pr 5297 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-sbc 3683 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-nul 4213 df-if 4416 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-br 5032 df-opab 5094 df-id 5430 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-res 5538 df-iota 6298 df-fun 6342 df-fv 6348 |
This theorem is referenced by: fntpb 6983 rabren3dioph 40201 nnsum4primesodd 44774 nnsum4primesoddALTV 44775 |
Copyright terms: Public domain | W3C validator |