MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fpr Structured version   Visualization version   GIF version

Theorem fpr 7173
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 6621 . . 3 (𝐴𝐵 → Fun {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩})
63, 4dmprop 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}))
85, 6, 7sylanblrc 590 . 2 (𝐴𝐵 → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} Fn {𝐴, 𝐵})
9 df-pr 4628 . . . . 5 {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩})
109rneqi 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⟩})
121rnsnop 6243 . . . . . 6 ran {⟨𝐴, 𝐶⟩} = {𝐶}
132rnsnop 6243 . . . . . 6 ran {⟨𝐵, 𝐷⟩} = {𝐷}
1412, 13uneq12i 4165 . . . . 5 (ran {⟨𝐴, 𝐶⟩} ∪ ran {⟨𝐵, 𝐷⟩}) = ({𝐶} ∪ {𝐷})
15 df-pr 4628 . . . . 5 {𝐶, 𝐷} = ({𝐶} ∪ {𝐷})
1614, 15eqtr4i 2767 . . . 4 (ran {⟨𝐴, 𝐶⟩} ∪ ran {⟨𝐵, 𝐷⟩}) = {𝐶, 𝐷}
1710, 11, 163eqtri 2768 . . 3 ran {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = {𝐶, 𝐷}
1817eqimssi 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}))
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 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
  Copyright terms: Public domain W3C validator