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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.) (Proof shortened by BJ, 26-Sep-2024.) |
| 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 | fvpr2.1 | . 2 ⊢ 𝐵 ∈ V | |
| 2 | fvpr2.2 | . 2 ⊢ 𝐷 ∈ V | |
| 3 | fvpr2g 7211 | . 2 ⊢ ((𝐵 ∈ V ∧ 𝐷 ∈ V ∧ 𝐴 ≠ 𝐵) → ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐵) = 𝐷) | |
| 4 | 1, 2, 3 | mp3an12 1453 | 1 ⊢ (𝐴 ≠ 𝐵 → ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}‘𝐵) = 𝐷) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 {cpr 4628 〈cop 4632 ‘cfv 6561 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-res 5697 df-iota 6514 df-fun 6563 df-fv 6569 |
| This theorem is referenced by: fprb 7214 fnprb 7228 m2detleiblem3 22635 m2detleiblem4 22636 axlowdimlem6 28962 umgr2v2evd2 29545 ex-fv 30462 bj-endcomp 37318 nnsum3primes4 47775 nnsum3primesgbe 47779 zlmodzxzldeplem3 48419 2arymaptfo 48575 prelrrx2b 48635 rrx2plordisom 48644 ehl2eudisval0 48646 itscnhlinecirc02p 48706 |
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