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Theorem fpr 5437
 Description: A function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.) (The proof was 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.3 . . . . . 6 C V
2 fpr.4 . . . . . 6 D V
31, 2funpr 5151 . . . . 5 (AB → Fun {A, C, B, D})
41, 2dmprop 5070 . . . . 5 dom {A, C, B, D} = {A, B}
53, 4jctir 524 . . . 4 (AB → (Fun {A, C, B, D} dom {A, C, B, D} = {A, B}))
6 df-fn 4790 . . . 4 ({A, C, B, D} Fn {A, B} ↔ (Fun {A, C, B, D} dom {A, C, B, D} = {A, B}))
75, 6sylibr 203 . . 3 (AB → {A, C, B, D} Fn {A, B})
8 fpr.1 . . . . . . 7 A V
98rnsnop 5075 . . . . . 6 ran {A, C} = {C}
10 fpr.2 . . . . . . 7 B V
1110rnsnop 5075 . . . . . 6 ran {B, D} = {D}
129, 11uneq12i 3416 . . . . 5 (ran {A, C} ∪ ran {B, D}) = ({C} ∪ {D})
13 df-pr 3742 . . . . . . 7 {A, C, B, D} = ({A, C} ∪ {B, D})
1413rneqi 4957 . . . . . 6 ran {A, C, B, D} = ran ({A, C} ∪ {B, D})
15 rnun 5036 . . . . . 6 ran ({A, C} ∪ {B, D}) = (ran {A, C} ∪ ran {B, D})
1614, 15eqtri 2373 . . . . 5 ran {A, C, B, D} = (ran {A, C} ∪ ran {B, D})
17 df-pr 3742 . . . . 5 {C, D} = ({C} ∪ {D})
1812, 16, 173eqtr4i 2383 . . . 4 ran {A, C, B, D} = {C, D}
1918eqimssi 3325 . . 3 ran {A, C, B, D} {C, D}
207, 19jctir 524 . 2 (AB → ({A, C, B, D} Fn {A, B} ran {A, C, B, D} {C, D}))
21 df-f 4791 . 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 203 1 (AB → {A, C, B, D}:{A, B}–→{C, D})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  Vcvv 2859   ∪ cun 3207   ⊆ wss 3257  {csn 3737  {cpr 3738  ⟨cop 4561  dom cdm 4772  ran crn 4773  Fun wfun 4775   Fn wfn 4776  –→wf 4777 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-f 4791 This theorem is referenced by: (None)
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