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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 5268 | . . . . 5 ⊢ (𝐴 ≠ 𝐵 → Fun {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}) |
6 | 3, 4 | dmprop 5103 | . . . . 5 ⊢ dom {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = {𝐴, 𝐵} |
7 | 5, 6 | jctir 313 | . . . 4 ⊢ (𝐴 ≠ 𝐵 → (Fun {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} ∧ dom {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = {𝐴, 𝐵})) |
8 | df-fn 5219 | . . . 4 ⊢ ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} Fn {𝐴, 𝐵} ↔ (Fun {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} ∧ dom {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = {𝐴, 𝐵})) | |
9 | 7, 8 | sylibr 134 | . . 3 ⊢ (𝐴 ≠ 𝐵 → {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} Fn {𝐴, 𝐵}) |
10 | df-pr 3599 | . . . . . 6 ⊢ {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉}) | |
11 | 10 | rneqi 4855 | . . . . 5 ⊢ ran {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = ran ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉}) |
12 | rnun 5037 | . . . . 5 ⊢ ran ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉}) = (ran {〈𝐴, 𝐶〉} ∪ ran {〈𝐵, 𝐷〉}) | |
13 | 1 | rnsnop 5109 | . . . . . . 7 ⊢ ran {〈𝐴, 𝐶〉} = {𝐶} |
14 | 2 | rnsnop 5109 | . . . . . . 7 ⊢ ran {〈𝐵, 𝐷〉} = {𝐷} |
15 | 13, 14 | uneq12i 3287 | . . . . . 6 ⊢ (ran {〈𝐴, 𝐶〉} ∪ ran {〈𝐵, 𝐷〉}) = ({𝐶} ∪ {𝐷}) |
16 | df-pr 3599 | . . . . . 6 ⊢ {𝐶, 𝐷} = ({𝐶} ∪ {𝐷}) | |
17 | 15, 16 | eqtr4i 2201 | . . . . 5 ⊢ (ran {〈𝐴, 𝐶〉} ∪ ran {〈𝐵, 𝐷〉}) = {𝐶, 𝐷} |
18 | 11, 12, 17 | 3eqtri 2202 | . . . 4 ⊢ ran {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = {𝐶, 𝐷} |
19 | 18 | eqimssi 3211 | . . 3 ⊢ ran {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} ⊆ {𝐶, 𝐷} |
20 | 9, 19 | jctir 313 | . 2 ⊢ (𝐴 ≠ 𝐵 → ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} Fn {𝐴, 𝐵} ∧ ran {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} ⊆ {𝐶, 𝐷})) |
21 | df-f 5220 | . 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 134 | 1 ⊢ (𝐴 ≠ 𝐵 → {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}:{𝐴, 𝐵}⟶{𝐶, 𝐷}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 ≠ wne 2347 Vcvv 2737 ∪ cun 3127 ⊆ wss 3129 {csn 3592 {cpr 3593 〈cop 3595 dom cdm 4626 ran crn 4627 Fun wfun 5210 Fn wfn 5211 ⟶wf 5212 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-br 4004 df-opab 4065 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-fun 5218 df-fn 5219 df-f 5220 |
This theorem is referenced by: (None) |
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