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| 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 470 | . 2 ⊢ (𝐴 ∈ V ∧ 𝐵 ∈ V) |
| 4 | funpr.3 | . . 3 ⊢ 𝐶 ∈ V | |
| 5 | funpr.4 | . . 3 ⊢ 𝐷 ∈ V | |
| 6 | 4, 5 | pm3.2i 470 | . 2 ⊢ (𝐶 ∈ V ∧ 𝐷 ∈ V) |
| 7 | funprg 6620 | . 2 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V) ∧ 𝐴 ≠ 𝐵) → Fun {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}) | |
| 8 | 3, 6, 7 | mp3an12 1453 | 1 ⊢ (𝐴 ≠ 𝐵 → Fun {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 {cpr 4628 〈cop 4632 Fun wfun 6555 |
| 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-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-sb 2065 df-mo 2540 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-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-fun 6563 |
| This theorem is referenced by: funtp 6623 fpr 7174 fnprb 7228 1sdomOLD 9285 |
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