![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > funpr | Structured version Visualization version GIF version |
Description: A function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.) |
Ref | Expression |
---|---|
funpr.1 | ⊢ 𝐴 ∈ V |
funpr.2 | ⊢ 𝐵 ∈ V |
funpr.3 | ⊢ 𝐶 ∈ V |
funpr.4 | ⊢ 𝐷 ∈ V |
Ref | Expression |
---|---|
funpr | ⊢ (𝐴 ≠ 𝐵 → Fun {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funpr.1 | . . 3 ⊢ 𝐴 ∈ V | |
2 | funpr.2 | . . 3 ⊢ 𝐵 ∈ V | |
3 | 1, 2 | pm3.2i 463 | . 2 ⊢ (𝐴 ∈ V ∧ 𝐵 ∈ V) |
4 | funpr.3 | . . 3 ⊢ 𝐶 ∈ V | |
5 | funpr.4 | . . 3 ⊢ 𝐷 ∈ V | |
6 | 4, 5 | pm3.2i 463 | . 2 ⊢ (𝐶 ∈ V ∧ 𝐷 ∈ V) |
7 | funprg 6155 | . 2 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V) ∧ 𝐴 ≠ 𝐵) → Fun {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}) | |
8 | 3, 6, 7 | mp3an12 1576 | 1 ⊢ (𝐴 ≠ 𝐵 → Fun {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 ∈ wcel 2157 ≠ wne 2972 Vcvv 3386 {cpr 4371 〈cop 4375 Fun wfun 6096 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 ax-sep 4976 ax-nul 4984 ax-pr 5098 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ne 2973 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3388 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-nul 4117 df-if 4279 df-sn 4370 df-pr 4372 df-op 4376 df-br 4845 df-opab 4907 df-id 5221 df-xp 5319 df-rel 5320 df-cnv 5321 df-co 5322 df-dm 5323 df-fun 6104 |
This theorem is referenced by: funtp 6158 fpr 6650 fnprb 6702 1sdom 8406 |
Copyright terms: Public domain | W3C validator |