Theorem funsn 5148

Description: A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 12-Aug-1994.) (Revised by Scott Fenton, 16-Apr-2021.)

Assertion

Ref	Expression
funsn	⊢ Fun {⟨A, B⟩}

Proof of Theorem funsn

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffun6 5125	. 2 ⊢ (Fun {⟨A, B⟩} ↔ ∀x∃*y x{⟨A, B⟩}y)
2		moeq 3013	. . . 4 ⊢ ∃*y y = B
3	2	a1i 10	. . 3 ⊢ (x = A → ∃*y y = B)
4		df-br 4641	. . . . . 6 ⊢ (x{⟨A, B⟩}y ↔ ⟨x, y⟩ ∈ {⟨A, B⟩})
5		vex 2863	. . . . . . . . 9 ⊢ x ∈ V
6		vex 2863	. . . . . . . . 9 ⊢ y ∈ V
7	5, 6	opex 4589	. . . . . . . 8 ⊢ ⟨x, y⟩ ∈ V
8	7	elsnc 3757	. . . . . . 7 ⊢ (⟨x, y⟩ ∈ {⟨A, B⟩} ↔ ⟨x, y⟩ = ⟨A, B⟩)
9		opth 4603	. . . . . . 7 ⊢ (⟨x, y⟩ = ⟨A, B⟩ ↔ (x = A ∧ y = B))
10	8, 9	bitri 240	. . . . . 6 ⊢ (⟨x, y⟩ ∈ {⟨A, B⟩} ↔ (x = A ∧ y = B))
11	4, 10	bitri 240	. . . . 5 ⊢ (x{⟨A, B⟩}y ↔ (x = A ∧ y = B))
12	11	mobii 2240	. . . 4 ⊢ (∃*y x{⟨A, B⟩}y ↔ ∃*y(x = A ∧ y = B))
13		moanimv 2262	. . . 4 ⊢ (∃*y(x = A ∧ y = B) ↔ (x = A → ∃*y y = B))
14	12, 13	bitri 240	. . 3 ⊢ (∃*y x{⟨A, B⟩}y ↔ (x = A → ∃*y y = B))
15	3, 14	mpbir 200	. 2 ⊢ ∃*y x{⟨A, B⟩}y
16	1, 15	mpgbir 1550	1 ⊢ Fun {⟨A, B⟩}

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃*wmo 2205  {csn 3738  ⟨cop 4562   class class class wbr 4640  Fun wfun 4776

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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-id 4768  df-cnv 4786  df-fun 4790

This theorem is referenced by:  funsngOLD  5149  funprg  5150  fnsn  5153  fvsn  5446