![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 7189 | . 2 ⊢ ((𝐵 ∈ V ∧ 𝐷 ∈ V ∧ 𝐴 ≠ 𝐵) → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐵) = 𝐷) | |
4 | 1, 2, 3 | mp3an12 1452 | 1 ⊢ (𝐴 ≠ 𝐵 → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐵) = 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 Vcvv 3475 {cpr 4631 ⟨cop 4635 ‘cfv 6544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-res 5689 df-iota 6496 df-fun 6546 df-fv 6552 |
This theorem is referenced by: fprb 7195 fnprb 7210 m2detleiblem3 22131 m2detleiblem4 22132 axlowdimlem6 28205 umgr2v2evd2 28784 ex-fv 29696 bj-endcomp 36198 nnsum3primes4 46456 nnsum3primesgbe 46460 zlmodzxzldeplem3 47183 2arymaptfo 47340 prelrrx2b 47400 rrx2plordisom 47409 ehl2eudisval0 47411 itscnhlinecirc02p 47471 |
Copyright terms: Public domain | W3C validator |