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Mirrors > Home > MPE Home > Th. List > fvpr2 | Structured version Visualization version GIF version |
Description: The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.) (Proof shortened by BJ, 26-Sep-2024.) |
Ref | Expression |
---|---|
fvpr2.1 | ⊢ 𝐵 ∈ V |
fvpr2.2 | ⊢ 𝐷 ∈ V |
Ref | Expression |
---|---|
fvpr2 | ⊢ (𝐴 ≠ 𝐵 → ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐵) = 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvpr2.1 | . 2 ⊢ 𝐵 ∈ V | |
2 | fvpr2.2 | . 2 ⊢ 𝐷 ∈ V | |
3 | fvpr2g 7133 | . 2 ⊢ ((𝐵 ∈ V ∧ 𝐷 ∈ V ∧ 𝐴 ≠ 𝐵) → ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐵) = 𝐷) | |
4 | 1, 2, 3 | mp3an12 1451 | 1 ⊢ (𝐴 ≠ 𝐵 → ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐵) = 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ≠ wne 2941 Vcvv 3443 {cpr 4586 〈cop 4590 ‘cfv 6493 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-res 5643 df-iota 6445 df-fun 6495 df-fv 6501 |
This theorem is referenced by: fprb 7139 fnprb 7154 m2detleiblem3 21962 m2detleiblem4 21963 axlowdimlem6 27782 umgr2v2evd2 28361 ex-fv 29273 bj-endcomp 35755 nnsum3primes4 45912 nnsum3primesgbe 45916 zlmodzxzldeplem3 46515 2arymaptfo 46672 prelrrx2b 46732 rrx2plordisom 46741 ehl2eudisval0 46743 itscnhlinecirc02p 46803 |
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