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Mirrors > Home > ILE Home > Th. List > funpr | 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 270 | . 2 ⊢ (𝐴 ∈ V ∧ 𝐵 ∈ V) |
4 | funpr.3 | . . 3 ⊢ 𝐶 ∈ V | |
5 | funpr.4 | . . 3 ⊢ 𝐷 ∈ V | |
6 | 4, 5 | pm3.2i 270 | . 2 ⊢ (𝐶 ∈ V ∧ 𝐷 ∈ V) |
7 | funprg 5248 | . 2 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V) ∧ 𝐴 ≠ 𝐵) → Fun {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}) | |
8 | 3, 6, 7 | mp3an12 1322 | 1 ⊢ (𝐴 ≠ 𝐵 → Fun {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 2141 ≠ wne 2340 Vcvv 2730 {cpr 3584 〈cop 3586 Fun wfun 5192 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-fun 5200 |
This theorem is referenced by: funtp 5251 fpr 5678 |
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