Theorem fpr 7088

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	⊢ (𝐴 ≠ 𝐵 → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}:{𝐴, 𝐵}⟶{𝐶, 𝐷})

Proof of Theorem fpr

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 6538	. . 3 ⊢ (𝐴 ≠ 𝐵 → Fun {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩})
6	3, 4	dmprop 6166	. . 3 ⊢ dom {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = {𝐴, 𝐵}
7		df-fn 6485	. . 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 4580	. . . . 5 ⊢ {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩})
10	9	rneqi 5879	. . . 4 ⊢ ran {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = ran ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩})
11		rnun 6094	. . . 4 ⊢ ran ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩}) = (ran {⟨𝐴, 𝐶⟩} ∪ ran {⟨𝐵, 𝐷⟩})
12	1	rnsnop 6173	. . . . . 6 ⊢ ran {⟨𝐴, 𝐶⟩} = {𝐶}
13	2	rnsnop 6173	. . . . . 6 ⊢ ran {⟨𝐵, 𝐷⟩} = {𝐷}
14	12, 13	uneq12i 4117	. . . . 5 ⊢ (ran {⟨𝐴, 𝐶⟩} ∪ ran {⟨𝐵, 𝐷⟩}) = ({𝐶} ∪ {𝐷})
15		df-pr 4580	. . . . 5 ⊢ {𝐶, 𝐷} = ({𝐶} ∪ {𝐷})
16	14, 15	eqtr4i 2755	. . . 4 ⊢ (ran {⟨𝐴, 𝐶⟩} ∪ ran {⟨𝐵, 𝐷⟩}) = {𝐶, 𝐷}
17	10, 11, 16	3eqtri 2756	. . 3 ⊢ ran {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = {𝐶, 𝐷}
18	17	eqimssi 3996	. 2 ⊢ ran {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} ⊆ {𝐶, 𝐷}
19		df-f 6486	. 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})

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  Vcvv 3436   ∪ cun 3901   ⊆ wss 3903  {csn 4577  {cpr 4579  ⟨cop 4583  dom cdm 5619  ran crn 5620  Fun wfun 6476   Fn wfn 6477  ⟶wf 6478

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5093  df-opab 5155  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-fun 6484  df-fn 6485  df-f 6486

This theorem is referenced by:  fprg  7089  fprb  7130  axlowdimlem4  28890  coinfliprv  34457  poimirlem22  37632  nnsum3primes4  47782  nnsum3primesgbe  47786