Theorem fpr 6900

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 6384	. . 3 ⊢ (𝐴 ≠ 𝐵 → Fun {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩})
6	3, 4	dmprop 6039	. . 3 ⊢ dom {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = {𝐴, 𝐵}
7		df-fn 6331	. . 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 594	. 2 ⊢ (𝐴 ≠ 𝐵 → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} Fn {𝐴, 𝐵})
9		df-pr 4518	. . . . 5 ⊢ {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩})
10	9	rneqi 5771	. . . 4 ⊢ ran {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = ran ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩})
11		rnun 5969	. . . 4 ⊢ ran ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩}) = (ran {⟨𝐴, 𝐶⟩} ∪ ran {⟨𝐵, 𝐷⟩})
12	1	rnsnop 6046	. . . . . 6 ⊢ ran {⟨𝐴, 𝐶⟩} = {𝐶}
13	2	rnsnop 6046	. . . . . 6 ⊢ ran {⟨𝐵, 𝐷⟩} = {𝐷}
14	12, 13	uneq12i 4062	. . . . 5 ⊢ (ran {⟨𝐴, 𝐶⟩} ∪ ran {⟨𝐵, 𝐷⟩}) = ({𝐶} ∪ {𝐷})
15		df-pr 4518	. . . . 5 ⊢ {𝐶, 𝐷} = ({𝐶} ∪ {𝐷})
16	14, 15	eqtr4i 2785	. . . 4 ⊢ (ran {⟨𝐴, 𝐶⟩} ∪ ran {⟨𝐵, 𝐷⟩}) = {𝐶, 𝐷}
17	10, 11, 16	3eqtri 2786	. . 3 ⊢ ran {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = {𝐶, 𝐷}
18	17	eqimssi 3946	. 2 ⊢ ran {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} ⊆ {𝐶, 𝐷}
19		df-f 6332	. 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 594	1 ⊢ (𝐴 ≠ 𝐵 → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}:{𝐴, 𝐵}⟶{𝐶, 𝐷})

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2112   ≠ wne 2949  Vcvv 3407   ∪ cun 3852   ⊆ wss 3854  {csn 4515  {cpr 4517  ⟨cop 4521  dom cdm 5517  ran crn 5518  Fun wfun 6322   Fn wfn 6323  ⟶wf 6324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5162  ax-nul 5169  ax-pr 5291

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2899  df-ne 2950  df-ral 3073  df-rex 3074  df-rab 3077  df-v 3409  df-dif 3857  df-un 3859  df-in 3861  df-ss 3871  df-nul 4222  df-if 4414  df-sn 4516  df-pr 4518  df-op 4522  df-br 5026  df-opab 5088  df-id 5423  df-xp 5523  df-rel 5524  df-cnv 5525  df-co 5526  df-dm 5527  df-rn 5528  df-fun 6330  df-fn 6331  df-f 6332

This theorem is referenced by:  fprg  6901  fprb  6940  1sdom  8735  axlowdimlem4  26823  coinfliprv  31953  poimirlem22  35344  nnsum3primes4  44658  nnsum3primesgbe  44662