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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 7207 | . 2 ⊢ ((𝐵 ∈ V ∧ 𝐷 ∈ V ∧ 𝐴 ≠ 𝐵) → ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐵) = 𝐷) | |
4 | 1, 2, 3 | mp3an12 1448 | 1 ⊢ (𝐴 ≠ 𝐵 → ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐵) = 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 Vcvv 3462 {cpr 4635 〈cop 4639 ‘cfv 6556 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5306 ax-nul 5313 ax-pr 5435 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4916 df-br 5156 df-opab 5218 df-id 5582 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-res 5696 df-iota 6508 df-fun 6558 df-fv 6564 |
This theorem is referenced by: fprb 7213 fnprb 7227 m2detleiblem3 22625 m2detleiblem4 22626 axlowdimlem6 28884 umgr2v2evd2 29467 ex-fv 30379 bj-endcomp 37026 nnsum3primes4 47378 nnsum3primesgbe 47382 zlmodzxzldeplem3 47903 2arymaptfo 48060 prelrrx2b 48120 rrx2plordisom 48129 ehl2eudisval0 48131 itscnhlinecirc02p 48191 |
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