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Theorem fpr 5699
Description: A function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Hypotheses
Ref Expression
fpr.1 𝐴 ∈ V
fpr.2 𝐵 ∈ V
fpr.3 𝐶 ∈ V
fpr.4 𝐷 ∈ V
Assertion
Ref Expression
fpr (𝐴𝐵 → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}:{𝐴, 𝐵}⟶{𝐶, 𝐷})

Proof of Theorem fpr
StepHypRef 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
51, 2, 3, 4funpr 5269 . . . . 5 (𝐴𝐵 → Fun {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩})
63, 4dmprop 5104 . . . . 5 dom {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = {𝐴, 𝐵}
75, 6jctir 313 . . . 4 (𝐴𝐵 → (Fun {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} ∧ dom {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = {𝐴, 𝐵}))
8 df-fn 5220 . . . 4 ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} Fn {𝐴, 𝐵} ↔ (Fun {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} ∧ dom {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = {𝐴, 𝐵}))
97, 8sylibr 134 . . 3 (𝐴𝐵 → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} Fn {𝐴, 𝐵})
10 df-pr 3600 . . . . . 6 {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩})
1110rneqi 4856 . . . . 5 ran {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = ran ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩})
12 rnun 5038 . . . . 5 ran ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩}) = (ran {⟨𝐴, 𝐶⟩} ∪ ran {⟨𝐵, 𝐷⟩})
131rnsnop 5110 . . . . . . 7 ran {⟨𝐴, 𝐶⟩} = {𝐶}
142rnsnop 5110 . . . . . . 7 ran {⟨𝐵, 𝐷⟩} = {𝐷}
1513, 14uneq12i 3288 . . . . . 6 (ran {⟨𝐴, 𝐶⟩} ∪ ran {⟨𝐵, 𝐷⟩}) = ({𝐶} ∪ {𝐷})
16 df-pr 3600 . . . . . 6 {𝐶, 𝐷} = ({𝐶} ∪ {𝐷})
1715, 16eqtr4i 2201 . . . . 5 (ran {⟨𝐴, 𝐶⟩} ∪ ran {⟨𝐵, 𝐷⟩}) = {𝐶, 𝐷}
1811, 12, 173eqtri 2202 . . . 4 ran {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = {𝐶, 𝐷}
1918eqimssi 3212 . . 3 ran {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} ⊆ {𝐶, 𝐷}
209, 19jctir 313 . 2 (𝐴𝐵 → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} Fn {𝐴, 𝐵} ∧ ran {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} ⊆ {𝐶, 𝐷}))
21 df-f 5221 . 2 ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}:{𝐴, 𝐵}⟶{𝐶, 𝐷} ↔ ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} Fn {𝐴, 𝐵} ∧ ran {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} ⊆ {𝐶, 𝐷}))
2220, 21sylibr 134 1 (𝐴𝐵 → {⟨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 2738  cun 3128  wss 3130  {csn 3593  {cpr 3594  cop 3596  dom cdm 4627  ran crn 4628  Fun wfun 5211   Fn wfn 5212  wf 5213
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 4122  ax-pow 4175  ax-pr 4210
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 2740  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-br 4005  df-opab 4066  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-fun 5219  df-fn 5220  df-f 5221
This theorem is referenced by: (None)
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