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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.) |
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 | prcom 4581 | . . 3 ⊢ {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = {〈𝐵, 𝐷〉, 〈𝐴, 𝐶〉} | |
2 | 1 | fveq1i 6546 | . 2 ⊢ ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐵) = ({〈𝐵, 𝐷〉, 〈𝐴, 𝐶〉}‘𝐵) |
3 | necom 3039 | . . 3 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴) | |
4 | fvpr2.1 | . . . 4 ⊢ 𝐵 ∈ V | |
5 | fvpr2.2 | . . . 4 ⊢ 𝐷 ∈ V | |
6 | 4, 5 | fvpr1 6826 | . . 3 ⊢ (𝐵 ≠ 𝐴 → ({〈𝐵, 𝐷〉, 〈𝐴, 𝐶〉}‘𝐵) = 𝐷) |
7 | 3, 6 | sylbi 218 | . 2 ⊢ (𝐴 ≠ 𝐵 → ({〈𝐵, 𝐷〉, 〈𝐴, 𝐶〉}‘𝐵) = 𝐷) |
8 | 2, 7 | syl5eq 2845 | 1 ⊢ (𝐴 ≠ 𝐵 → ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐵) = 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1525 ∈ wcel 2083 ≠ wne 2986 Vcvv 3440 {cpr 4480 〈cop 4484 ‘cfv 6232 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-ral 3112 df-rex 3113 df-rab 3116 df-v 3442 df-sbc 3712 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-sn 4479 df-pr 4481 df-op 4485 df-uni 4752 df-br 4969 df-opab 5031 df-id 5355 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-res 5462 df-iota 6196 df-fun 6234 df-fv 6240 |
This theorem is referenced by: fprb 6830 fnprb 6844 m2detleiblem3 20926 m2detleiblem4 20927 axlowdimlem6 26420 umgr2v2evd2 26996 ex-fv 27910 nnsum3primes4 43457 nnsum3primesgbe 43461 zlmodzxzldeplem3 44059 prelrrx2b 44204 rrx2plordisom 44213 ehl2eudisval0 44215 itscnhlinecirc02p 44275 |
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