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Mirrors > Home > ILE Home > Th. List > fpr | GIF version |
Description: A function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Ref | Expression |
---|---|
fpr.1 | ⊢ 𝐴 ∈ V |
fpr.2 | ⊢ 𝐵 ∈ V |
fpr.3 | ⊢ 𝐶 ∈ V |
fpr.4 | ⊢ 𝐷 ∈ V |
Ref | Expression |
---|---|
fpr | ⊢ (𝐴 ≠ 𝐵 → {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}:{𝐴, 𝐵}⟶{𝐶, 𝐷}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fpr.1 | . . . . . 6 ⊢ 𝐴 ∈ V | |
2 | fpr.2 | . . . . . 6 ⊢ 𝐵 ∈ V | |
3 | fpr.3 | . . . . . 6 ⊢ 𝐶 ∈ V | |
4 | fpr.4 | . . . . . 6 ⊢ 𝐷 ∈ V | |
5 | 1, 2, 3, 4 | funpr 5175 | . . . . 5 ⊢ (𝐴 ≠ 𝐵 → Fun {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}) |
6 | 3, 4 | dmprop 5013 | . . . . 5 ⊢ dom {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = {𝐴, 𝐵} |
7 | 5, 6 | jctir 311 | . . . 4 ⊢ (𝐴 ≠ 𝐵 → (Fun {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} ∧ dom {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = {𝐴, 𝐵})) |
8 | df-fn 5126 | . . . 4 ⊢ ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} Fn {𝐴, 𝐵} ↔ (Fun {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} ∧ dom {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = {𝐴, 𝐵})) | |
9 | 7, 8 | sylibr 133 | . . 3 ⊢ (𝐴 ≠ 𝐵 → {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} Fn {𝐴, 𝐵}) |
10 | df-pr 3534 | . . . . . 6 ⊢ {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉}) | |
11 | 10 | rneqi 4767 | . . . . 5 ⊢ ran {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = ran ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉}) |
12 | rnun 4947 | . . . . 5 ⊢ ran ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉}) = (ran {〈𝐴, 𝐶〉} ∪ ran {〈𝐵, 𝐷〉}) | |
13 | 1 | rnsnop 5019 | . . . . . . 7 ⊢ ran {〈𝐴, 𝐶〉} = {𝐶} |
14 | 2 | rnsnop 5019 | . . . . . . 7 ⊢ ran {〈𝐵, 𝐷〉} = {𝐷} |
15 | 13, 14 | uneq12i 3228 | . . . . . 6 ⊢ (ran {〈𝐴, 𝐶〉} ∪ ran {〈𝐵, 𝐷〉}) = ({𝐶} ∪ {𝐷}) |
16 | df-pr 3534 | . . . . . 6 ⊢ {𝐶, 𝐷} = ({𝐶} ∪ {𝐷}) | |
17 | 15, 16 | eqtr4i 2163 | . . . . 5 ⊢ (ran {〈𝐴, 𝐶〉} ∪ ran {〈𝐵, 𝐷〉}) = {𝐶, 𝐷} |
18 | 11, 12, 17 | 3eqtri 2164 | . . . 4 ⊢ ran {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = {𝐶, 𝐷} |
19 | 18 | eqimssi 3153 | . . 3 ⊢ ran {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} ⊆ {𝐶, 𝐷} |
20 | 9, 19 | jctir 311 | . 2 ⊢ (𝐴 ≠ 𝐵 → ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} Fn {𝐴, 𝐵} ∧ ran {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} ⊆ {𝐶, 𝐷})) |
21 | df-f 5127 | . 2 ⊢ ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}:{𝐴, 𝐵}⟶{𝐶, 𝐷} ↔ ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} Fn {𝐴, 𝐵} ∧ ran {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} ⊆ {𝐶, 𝐷})) | |
22 | 20, 21 | sylibr 133 | 1 ⊢ (𝐴 ≠ 𝐵 → {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}:{𝐴, 𝐵}⟶{𝐶, 𝐷}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 ≠ wne 2308 Vcvv 2686 ∪ cun 3069 ⊆ wss 3071 {csn 3527 {cpr 3528 〈cop 3530 dom cdm 4539 ran crn 4540 Fun wfun 5117 Fn wfn 5118 ⟶wf 5119 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-fun 5125 df-fn 5126 df-f 5127 |
This theorem is referenced by: (None) |
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