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Mirrors > Home > MPE Home > Th. List > fpr | Structured version Visualization version 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 5858 | . . . . 5 ⊢ (𝐴 ≠ 𝐵 → Fun {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}) |
6 | 3, 4 | dmprop 5528 | . . . . 5 ⊢ dom {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = {𝐴, 𝐵} |
7 | 5, 6 | jctir 559 | . . . 4 ⊢ (𝐴 ≠ 𝐵 → (Fun {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} ∧ dom {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = {𝐴, 𝐵})) |
8 | df-fn 5807 | . . . 4 ⊢ ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} Fn {𝐴, 𝐵} ↔ (Fun {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} ∧ dom {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = {𝐴, 𝐵})) | |
9 | 7, 8 | sylibr 223 | . . 3 ⊢ (𝐴 ≠ 𝐵 → {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} Fn {𝐴, 𝐵}) |
10 | df-pr 4128 | . . . . . 6 ⊢ {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉}) | |
11 | 10 | rneqi 5273 | . . . . 5 ⊢ ran {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = ran ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉}) |
12 | rnun 5460 | . . . . 5 ⊢ ran ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉}) = (ran {〈𝐴, 𝐶〉} ∪ ran {〈𝐵, 𝐷〉}) | |
13 | 1 | rnsnop 5534 | . . . . . . 7 ⊢ ran {〈𝐴, 𝐶〉} = {𝐶} |
14 | 2 | rnsnop 5534 | . . . . . . 7 ⊢ ran {〈𝐵, 𝐷〉} = {𝐷} |
15 | 13, 14 | uneq12i 3727 | . . . . . 6 ⊢ (ran {〈𝐴, 𝐶〉} ∪ ran {〈𝐵, 𝐷〉}) = ({𝐶} ∪ {𝐷}) |
16 | df-pr 4128 | . . . . . 6 ⊢ {𝐶, 𝐷} = ({𝐶} ∪ {𝐷}) | |
17 | 15, 16 | eqtr4i 2635 | . . . . 5 ⊢ (ran {〈𝐴, 𝐶〉} ∪ ran {〈𝐵, 𝐷〉}) = {𝐶, 𝐷} |
18 | 11, 12, 17 | 3eqtri 2636 | . . . 4 ⊢ ran {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = {𝐶, 𝐷} |
19 | 18 | eqimssi 3622 | . . 3 ⊢ ran {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} ⊆ {𝐶, 𝐷} |
20 | 9, 19 | jctir 559 | . 2 ⊢ (𝐴 ≠ 𝐵 → ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} Fn {𝐴, 𝐵} ∧ ran {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} ⊆ {𝐶, 𝐷})) |
21 | df-f 5808 | . 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 223 | 1 ⊢ (𝐴 ≠ 𝐵 → {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}:{𝐴, 𝐵}⟶{𝐶, 𝐷}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 ∪ cun 3538 ⊆ wss 3540 {csn 4125 {cpr 4127 〈cop 4131 dom cdm 5038 ran crn 5039 Fun wfun 5798 Fn wfn 5799 ⟶wf 5800 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-fun 5806 df-fn 5807 df-f 5808 |
This theorem is referenced by: fprg 6327 1sdom 8048 axlowdimlem4 25625 wlkntrllem1 26089 wlkntrllem3 26091 coinfliprv 29871 fprb 30916 poimirlem22 32601 nnsum3primes4 40204 nnsum3primesgbe 40208 |
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