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Theorem fpr 6908
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 . . . 4 𝐴 ∈ V
2 fpr.2 . . . 4 𝐵 ∈ V
3 fpr.3 . . . 4 𝐶 ∈ V
4 fpr.4 . . . 4 𝐷 ∈ V
51, 2, 3, 4funpr 6403 . . 3 (𝐴𝐵 → Fun {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩})
63, 4dmprop 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}))
85, 6, 7sylanblrc 590 . 2 (𝐴𝐵 → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} Fn {𝐴, 𝐵})
9 df-pr 4560 . . . . 5 {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩})
109rneqi 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⟩})
121rnsnop 6074 . . . . . 6 ran {⟨𝐴, 𝐶⟩} = {𝐶}
132rnsnop 6074 . . . . . 6 ran {⟨𝐵, 𝐷⟩} = {𝐷}
1412, 13uneq12i 4134 . . . . 5 (ran {⟨𝐴, 𝐶⟩} ∪ ran {⟨𝐵, 𝐷⟩}) = ({𝐶} ∪ {𝐷})
15 df-pr 4560 . . . . 5 {𝐶, 𝐷} = ({𝐶} ∪ {𝐷})
1614, 15eqtr4i 2844 . . . 4 (ran {⟨𝐴, 𝐶⟩} ∪ ran {⟨𝐵, 𝐷⟩}) = {𝐶, 𝐷}
1710, 11, 163eqtri 2845 . . 3 ran {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = {𝐶, 𝐷}
1817eqimssi 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}))
208, 18, 19sylanblrc 590 1 (𝐴𝐵 → {⟨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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