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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 | . . . 4 ⊢ 𝐴 ∈ V | |
2 | fpr.2 | . . . 4 ⊢ 𝐵 ∈ V | |
3 | fpr.3 | . . . 4 ⊢ 𝐶 ∈ V | |
4 | fpr.4 | . . . 4 ⊢ 𝐷 ∈ V | |
5 | 1, 2, 3, 4 | funpr 6403 | . . 3 ⊢ (𝐴 ≠ 𝐵 → Fun {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}) |
6 | 3, 4 | dmprop 6067 | . . 3 ⊢ dom {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = {𝐴, 𝐵} |
7 | df-fn 6351 | . . 3 ⊢ ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} Fn {𝐴, 𝐵} ↔ (Fun {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} ∧ dom {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = {𝐴, 𝐵})) | |
8 | 5, 6, 7 | sylanblrc 590 | . 2 ⊢ (𝐴 ≠ 𝐵 → {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} Fn {𝐴, 𝐵}) |
9 | df-pr 4560 | . . . . 5 ⊢ {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉}) | |
10 | 9 | rneqi 5800 | . . . 4 ⊢ ran {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = ran ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉}) |
11 | rnun 5997 | . . . 4 ⊢ ran ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉}) = (ran {〈𝐴, 𝐶〉} ∪ ran {〈𝐵, 𝐷〉}) | |
12 | 1 | rnsnop 6074 | . . . . . 6 ⊢ ran {〈𝐴, 𝐶〉} = {𝐶} |
13 | 2 | rnsnop 6074 | . . . . . 6 ⊢ ran {〈𝐵, 𝐷〉} = {𝐷} |
14 | 12, 13 | uneq12i 4134 | . . . . 5 ⊢ (ran {〈𝐴, 𝐶〉} ∪ ran {〈𝐵, 𝐷〉}) = ({𝐶} ∪ {𝐷}) |
15 | df-pr 4560 | . . . . 5 ⊢ {𝐶, 𝐷} = ({𝐶} ∪ {𝐷}) | |
16 | 14, 15 | eqtr4i 2844 | . . . 4 ⊢ (ran {〈𝐴, 𝐶〉} ∪ ran {〈𝐵, 𝐷〉}) = {𝐶, 𝐷} |
17 | 10, 11, 16 | 3eqtri 2845 | . . 3 ⊢ ran {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = {𝐶, 𝐷} |
18 | 17 | eqimssi 4022 | . 2 ⊢ ran {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} ⊆ {𝐶, 𝐷} |
19 | df-f 6352 | . 2 ⊢ ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}:{𝐴, 𝐵}⟶{𝐶, 𝐷} ↔ ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} Fn {𝐴, 𝐵} ∧ ran {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} ⊆ {𝐶, 𝐷})) | |
20 | 8, 18, 19 | sylanblrc 590 | 1 ⊢ (𝐴 ≠ 𝐵 → {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}:{𝐴, 𝐵}⟶{𝐶, 𝐷}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 Vcvv 3492 ∪ cun 3931 ⊆ wss 3933 {csn 4557 {cpr 4559 〈cop 4563 dom cdm 5548 ran crn 5549 Fun wfun 6342 Fn wfn 6343 ⟶wf 6344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-fun 6350 df-fn 6351 df-f 6352 |
This theorem is referenced by: fprg 6909 fprb 6948 1sdom 8709 axlowdimlem4 26658 coinfliprv 31639 poimirlem22 34795 nnsum3primes4 43830 nnsum3primesgbe 43834 |
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