![]() |
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 7137 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐶 ∈ V ∧ 𝐴 ≠ 𝐵) → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐴) = 𝐶) | |
4 | 1, 2, 3 | mp3an12 1452 | 1 ⊢ (𝐴 ≠ 𝐵 → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐴) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 Vcvv 3444 {cpr 4589 ⟨cop 4593 ‘cfv 6497 |
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 5257 ax-nul 5264 ax-pr 5385 |
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 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-res 5646 df-iota 6449 df-fun 6499 df-fv 6505 |
This theorem is referenced by: fvpr2OLD 7143 fprb 7144 fvtp1 7145 fnprb 7159 m2detleiblem3 21994 m2detleiblem4 21995 axlowdimlem6 27938 umgr2v2evd2 28517 bj-endbase 35833 poimirlem22 36146 nnsum3primes4 46066 nnsum3primesgbe 46070 zlmodzxzldeplem3 46669 zlmodzxzldeplem4 46670 2arymaptfo 46826 prelrrx2b 46886 rrx2plordisom 46895 ehl2eudisval0 46897 line2 46924 itscnhlinecirc02p 46957 |
Copyright terms: Public domain | W3C validator |