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Mirrors > Home > MPE Home > Th. List > fvpr1g | Structured version Visualization version GIF version |
Description: The value of a function with a domain of (at most) two elements. (Contributed by Alexander van der Vekens, 3-Dec-2017.) |
Ref | Expression |
---|---|
fvpr1g | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐴) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pr 4633 | . . . . 5 ⊢ {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉}) | |
2 | 1 | fveq1i 6897 | . . . 4 ⊢ ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐴) = (({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉})‘𝐴) |
3 | necom 2983 | . . . . 5 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴) | |
4 | fvunsn 7188 | . . . . 5 ⊢ (𝐵 ≠ 𝐴 → (({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉})‘𝐴) = ({〈𝐴, 𝐶〉}‘𝐴)) | |
5 | 3, 4 | sylbi 216 | . . . 4 ⊢ (𝐴 ≠ 𝐵 → (({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉})‘𝐴) = ({〈𝐴, 𝐶〉}‘𝐴)) |
6 | 2, 5 | eqtrid 2777 | . . 3 ⊢ (𝐴 ≠ 𝐵 → ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐴) = ({〈𝐴, 𝐶〉}‘𝐴)) |
7 | 6 | 3ad2ant3 1132 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐴) = ({〈𝐴, 𝐶〉}‘𝐴)) |
8 | fvsng 7189 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ({〈𝐴, 𝐶〉}‘𝐴) = 𝐶) | |
9 | 8 | 3adant3 1129 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({〈𝐴, 𝐶〉}‘𝐴) = 𝐶) |
10 | 7, 9 | eqtrd 2765 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐴) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ∪ cun 3942 {csn 4630 {cpr 4632 〈cop 4636 ‘cfv 6549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-res 5690 df-iota 6501 df-fun 6551 df-fv 6557 |
This theorem is referenced by: fvpr2g 7200 fvpr1 7202 fvtp1g 7210 fpropnf1 7277 f1prex 7293 wrdlen2i 14929 fvpr0o 17544 linds2eq 33193 zlmodzxzscm 47604 zlmodzxzadd 47605 lincvalpr 47669 ldepspr 47724 2arymptfv 47906 fv1prop 47955 prelrrx2b 47970 line2ylem 48007 line2 48008 line2x 48010 line2y 48011 |
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