![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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.) (Proof shortened by BJ, 26-Sep-2024.) |
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 | fvpr1.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | fvpr1.2 | . 2 ⊢ 𝐶 ∈ V | |
3 | fvpr1g 7199 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐶 ∈ V ∧ 𝐴 ≠ 𝐵) → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐴) = 𝐶) | |
4 | 1, 2, 3 | mp3an12 1448 | 1 ⊢ (𝐴 ≠ 𝐵 → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐴) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 Vcvv 3471 {cpr 4631 ⟨cop 4635 ‘cfv 6548 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-res 5690 df-iota 6500 df-fun 6550 df-fv 6556 |
This theorem is referenced by: fvpr2OLD 7205 fprb 7206 fvtp1 7207 fnprb 7220 m2detleiblem3 22530 m2detleiblem4 22531 axlowdimlem6 28757 umgr2v2evd2 29340 bj-endbase 36795 poimirlem22 37115 nnsum3primes4 47128 nnsum3primesgbe 47132 zlmodzxzldeplem3 47570 zlmodzxzldeplem4 47571 2arymaptfo 47727 prelrrx2b 47787 rrx2plordisom 47796 ehl2eudisval0 47798 line2 47825 itscnhlinecirc02p 47858 |
Copyright terms: Public domain | W3C validator |