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Mirrors > Home > MPE Home > Th. List > fvpr1 | 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 |
---|---|
fvpr1.1 | ⊢ 𝐴 ∈ V |
fvpr1.2 | ⊢ 𝐶 ∈ V |
Ref | Expression |
---|---|
fvpr1 | ⊢ (𝐴 ≠ 𝐵 → ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐴) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pr 4563 | . . . 4 ⊢ {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉}) | |
2 | 1 | fveq1i 6665 | . . 3 ⊢ ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐴) = (({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉})‘𝐴) |
3 | necom 3069 | . . . 4 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴) | |
4 | fvunsn 6935 | . . . 4 ⊢ (𝐵 ≠ 𝐴 → (({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉})‘𝐴) = ({〈𝐴, 𝐶〉}‘𝐴)) | |
5 | 3, 4 | sylbi 219 | . . 3 ⊢ (𝐴 ≠ 𝐵 → (({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉})‘𝐴) = ({〈𝐴, 𝐶〉}‘𝐴)) |
6 | 2, 5 | syl5eq 2868 | . 2 ⊢ (𝐴 ≠ 𝐵 → ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐴) = ({〈𝐴, 𝐶〉}‘𝐴)) |
7 | fvpr1.1 | . . 3 ⊢ 𝐴 ∈ V | |
8 | fvpr1.2 | . . 3 ⊢ 𝐶 ∈ V | |
9 | 7, 8 | fvsn 6937 | . 2 ⊢ ({〈𝐴, 𝐶〉}‘𝐴) = 𝐶 |
10 | 6, 9 | syl6eq 2872 | 1 ⊢ (𝐴 ≠ 𝐵 → ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐴) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 Vcvv 3494 ∪ cun 3933 {csn 4560 {cpr 4562 〈cop 4566 ‘cfv 6349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-res 5561 df-iota 6308 df-fun 6351 df-fv 6357 |
This theorem is referenced by: fvpr2 6947 fprb 6950 fvtp1 6951 fnprb 6965 m2detleiblem3 21232 m2detleiblem4 21233 axlowdimlem6 26727 umgr2v2evd2 27303 bj-endbase 34591 poimirlem22 34908 nnsum3primes4 43947 nnsum3primesgbe 43951 zlmodzxzldeplem3 44551 zlmodzxzldeplem4 44552 prelrrx2b 44695 rrx2plordisom 44704 ehl2eudisval0 44706 line2 44733 itscnhlinecirc02p 44766 |
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