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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 4670 | . . 3 ⊢ {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = {〈𝐵, 𝐷〉, 〈𝐴, 𝐶〉} | |
2 | 1 | fveq1i 6673 | . 2 ⊢ ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐵) = ({〈𝐵, 𝐷〉, 〈𝐴, 𝐶〉}‘𝐵) |
3 | necom 3071 | . . 3 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴) | |
4 | fvpr2.1 | . . . 4 ⊢ 𝐵 ∈ V | |
5 | fvpr2.2 | . . . 4 ⊢ 𝐷 ∈ V | |
6 | 4, 5 | fvpr1 6954 | . . 3 ⊢ (𝐵 ≠ 𝐴 → ({〈𝐵, 𝐷〉, 〈𝐴, 𝐶〉}‘𝐵) = 𝐷) |
7 | 3, 6 | sylbi 219 | . 2 ⊢ (𝐴 ≠ 𝐵 → ({〈𝐵, 𝐷〉, 〈𝐴, 𝐶〉}‘𝐵) = 𝐷) |
8 | 2, 7 | syl5eq 2870 | 1 ⊢ (𝐴 ≠ 𝐵 → ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐵) = 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 Vcvv 3496 {cpr 4571 〈cop 4575 ‘cfv 6357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-res 5569 df-iota 6316 df-fun 6359 df-fv 6365 |
This theorem is referenced by: fprb 6958 fnprb 6973 m2detleiblem3 21240 m2detleiblem4 21241 axlowdimlem6 26735 umgr2v2evd2 27311 ex-fv 28224 bj-endcomp 34600 nnsum3primes4 43960 nnsum3primesgbe 43964 zlmodzxzldeplem3 44564 prelrrx2b 44708 rrx2plordisom 44717 ehl2eudisval0 44719 itscnhlinecirc02p 44779 |
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