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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 4951 | . . . . 5 ⊢ (𝐴 ≠ 𝐵 → Fun {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}) |
6 | 3, 4 | dmprop 4795 | . . . . 5 ⊢ dom {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = {𝐴, 𝐵} |
7 | 5, 6 | jctir 296 | . . . 4 ⊢ (𝐴 ≠ 𝐵 → (Fun {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} ∧ dom {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = {𝐴, 𝐵})) |
8 | df-fn 4905 | . . . 4 ⊢ ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} Fn {𝐴, 𝐵} ↔ (Fun {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} ∧ dom {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = {𝐴, 𝐵})) | |
9 | 7, 8 | sylibr 137 | . . 3 ⊢ (𝐴 ≠ 𝐵 → {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} Fn {𝐴, 𝐵}) |
10 | df-pr 3382 | . . . . . 6 ⊢ {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉}) | |
11 | 10 | rneqi 4562 | . . . . 5 ⊢ ran {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = ran ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉}) |
12 | rnun 4732 | . . . . 5 ⊢ ran ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉}) = (ran {〈𝐴, 𝐶〉} ∪ ran {〈𝐵, 𝐷〉}) | |
13 | 1 | rnsnop 4801 | . . . . . . 7 ⊢ ran {〈𝐴, 𝐶〉} = {𝐶} |
14 | 2 | rnsnop 4801 | . . . . . . 7 ⊢ ran {〈𝐵, 𝐷〉} = {𝐷} |
15 | 13, 14 | uneq12i 3095 | . . . . . 6 ⊢ (ran {〈𝐴, 𝐶〉} ∪ ran {〈𝐵, 𝐷〉}) = ({𝐶} ∪ {𝐷}) |
16 | df-pr 3382 | . . . . . 6 ⊢ {𝐶, 𝐷} = ({𝐶} ∪ {𝐷}) | |
17 | 15, 16 | eqtr4i 2063 | . . . . 5 ⊢ (ran {〈𝐴, 𝐶〉} ∪ ran {〈𝐵, 𝐷〉}) = {𝐶, 𝐷} |
18 | 11, 12, 17 | 3eqtri 2064 | . . . 4 ⊢ ran {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = {𝐶, 𝐷} |
19 | 18 | eqimssi 2999 | . . 3 ⊢ ran {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} ⊆ {𝐶, 𝐷} |
20 | 9, 19 | jctir 296 | . 2 ⊢ (𝐴 ≠ 𝐵 → ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} Fn {𝐴, 𝐵} ∧ ran {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} ⊆ {𝐶, 𝐷})) |
21 | df-f 4906 | . 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 137 | 1 ⊢ (𝐴 ≠ 𝐵 → {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}:{𝐴, 𝐵}⟶{𝐶, 𝐷}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∈ wcel 1393 ≠ wne 2204 Vcvv 2557 ∪ cun 2915 ⊆ wss 2917 {csn 3375 {cpr 3376 〈cop 3378 dom cdm 4345 ran crn 4346 Fun wfun 4896 Fn wfn 4897 ⟶wf 4898 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-fun 4904 df-fn 4905 df-f 4906 |
This theorem is referenced by: (None) |
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