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

Proof of Theorem fpr

Step	Hyp	Ref	Expression
1		fpr.3	. . . . . 6 ⊢ C ∈ V
2		fpr.4	. . . . . 6 ⊢ D ∈ V
3	1, 2	funpr 5151	. . . . 5 ⊢ (A ≠ B → Fun {⟨A, C⟩, ⟨B, D⟩})
4	1, 2	dmprop 5070	. . . . 5 ⊢ dom {⟨A, C⟩, ⟨B, D⟩} = {A, B}
5	3, 4	jctir 524	. . . 4 ⊢ (A ≠ B → (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}))
7	5, 6	sylibr 203	. . 3 ⊢ (A ≠ B → {⟨A, C⟩, ⟨B, D⟩} Fn {A, B})
8		fpr.1	. . . . . . 7 ⊢ A ∈ V
9	8	rnsnop 5075	. . . . . 6 ⊢ ran {⟨A, C⟩} = {C}
10		fpr.2	. . . . . . 7 ⊢ B ∈ V
11	10	rnsnop 5075	. . . . . 6 ⊢ ran {⟨B, D⟩} = {D}
12	9, 11	uneq12i 3416	. . . . 5 ⊢ (ran {⟨A, C⟩} ∪ ran {⟨B, D⟩}) = ({C} ∪ {D})
13		df-pr 3742	. . . . . . 7 ⊢ {⟨A, C⟩, ⟨B, D⟩} = ({⟨A, C⟩} ∪ {⟨B, D⟩})
14	13	rneqi 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⟩})
16	14, 15	eqtri 2373	. . . . 5 ⊢ ran {⟨A, C⟩, ⟨B, D⟩} = (ran {⟨A, C⟩} ∪ ran {⟨B, D⟩})
17		df-pr 3742	. . . . 5 ⊢ {C, D} = ({C} ∪ {D})
18	12, 16, 17	3eqtr4i 2383	. . . 4 ⊢ ran {⟨A, C⟩, ⟨B, D⟩} = {C, D}
19	18	eqimssi 3325	. . 3 ⊢ ran {⟨A, C⟩, ⟨B, D⟩} ⊆ {C, D}
20	7, 19	jctir 524	. 2 ⊢ (A ≠ B → ({⟨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}))
22	20, 21	sylibr 203	1 ⊢ (A ≠ B → {⟨A, C⟩, ⟨B, D⟩}:{A, B}–→{C, D})

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-mp 5  ax-1 6  ax-2 7  ax-3 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)