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| 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 6621 | . . 3 ⊢ (𝐴 ≠ 𝐵 → Fun {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}) | 
| 6 | 3, 4 | dmprop 6236 | . . 3 ⊢ dom {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = {𝐴, 𝐵} | 
| 7 | df-fn 6563 | . . 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 4628 | . . . . 5 ⊢ {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉}) | |
| 10 | 9 | rneqi 5947 | . . . 4 ⊢ ran {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = ran ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉}) | 
| 11 | rnun 6164 | . . . 4 ⊢ ran ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉}) = (ran {〈𝐴, 𝐶〉} ∪ ran {〈𝐵, 𝐷〉}) | |
| 12 | 1 | rnsnop 6243 | . . . . . 6 ⊢ ran {〈𝐴, 𝐶〉} = {𝐶} | 
| 13 | 2 | rnsnop 6243 | . . . . . 6 ⊢ ran {〈𝐵, 𝐷〉} = {𝐷} | 
| 14 | 12, 13 | uneq12i 4165 | . . . . 5 ⊢ (ran {〈𝐴, 𝐶〉} ∪ ran {〈𝐵, 𝐷〉}) = ({𝐶} ∪ {𝐷}) | 
| 15 | df-pr 4628 | . . . . 5 ⊢ {𝐶, 𝐷} = ({𝐶} ∪ {𝐷}) | |
| 16 | 14, 15 | eqtr4i 2767 | . . . 4 ⊢ (ran {〈𝐴, 𝐶〉} ∪ ran {〈𝐵, 𝐷〉}) = {𝐶, 𝐷} | 
| 17 | 10, 11, 16 | 3eqtri 2768 | . . 3 ⊢ ran {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = {𝐶, 𝐷} | 
| 18 | 17 | eqimssi 4043 | . 2 ⊢ ran {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} ⊆ {𝐶, 𝐷} | 
| 19 | df-f 6564 | . 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 1539 ∈ wcel 2107 ≠ wne 2939 Vcvv 3479 ∪ cun 3948 ⊆ wss 3950 {csn 4625 {cpr 4627 〈cop 4631 dom cdm 5684 ran crn 5685 Fun wfun 6554 Fn wfn 6555 ⟶wf 6556 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-fun 6562 df-fn 6563 df-f 6564 | 
| This theorem is referenced by: fprg 7174 fprb 7215 1sdomOLD 9286 axlowdimlem4 28961 coinfliprv 34486 poimirlem22 37650 nnsum3primes4 47780 nnsum3primesgbe 47784 | 
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