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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 6907 | . . . 4 ⊢ ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐴) = (({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉})‘𝐴) |
3 | necom 2991 | . . . . 5 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴) | |
4 | fvunsn 7198 | . . . . 5 ⊢ (𝐵 ≠ 𝐴 → (({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉})‘𝐴) = ({〈𝐴, 𝐶〉}‘𝐴)) | |
5 | 3, 4 | sylbi 217 | . . . 4 ⊢ (𝐴 ≠ 𝐵 → (({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉})‘𝐴) = ({〈𝐴, 𝐶〉}‘𝐴)) |
6 | 2, 5 | eqtrid 2786 | . . 3 ⊢ (𝐴 ≠ 𝐵 → ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐴) = ({〈𝐴, 𝐶〉}‘𝐴)) |
7 | 6 | 3ad2ant3 1134 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐴) = ({〈𝐴, 𝐶〉}‘𝐴)) |
8 | fvsng 7199 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ({〈𝐴, 𝐶〉}‘𝐴) = 𝐶) | |
9 | 8 | 3adant3 1131 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({〈𝐴, 𝐶〉}‘𝐴) = 𝐶) |
10 | 7, 9 | eqtrd 2774 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐴) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∪ cun 3960 {csn 4630 {cpr 4632 〈cop 4636 ‘cfv 6562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-res 5700 df-iota 6515 df-fun 6564 df-fv 6570 |
This theorem is referenced by: fvpr2g 7210 fvpr1 7211 fvtp1g 7217 fpropnf1 7286 f1prex 7303 wrdlen2i 14977 fvpr0o 17605 linds2eq 33388 zlmodzxzscm 48201 zlmodzxzadd 48202 lincvalpr 48263 ldepspr 48318 2arymptfv 48499 fv1prop 48548 prelrrx2b 48563 line2ylem 48600 line2 48601 line2x 48603 line2y 48604 |
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