Theorem dmsnopg 5067

Description: The domain of a singleton of an ordered pair is the singleton of the first member. (Contributed by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
dmsnopg	⊢ (B ∈ V → dom {⟨A, B⟩} = {A})

Proof of Theorem dmsnopg

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

Step	Hyp	Ref	Expression
1		opeq2 4580	. . . . 5 ⊢ (y = B → ⟨A, y⟩ = ⟨A, B⟩)
2	1	sneqd 3747	. . . 4 ⊢ (y = B → {⟨A, y⟩} = {⟨A, B⟩})
3	2	dmeqd 4910	. . 3 ⊢ (y = B → dom {⟨A, y⟩} = dom {⟨A, B⟩})
4	3	eqeq1d 2361	. 2 ⊢ (y = B → (dom {⟨A, y⟩} = {A} ↔ dom {⟨A, B⟩} = {A}))
5		df-br 4641	. . . . . . 7 ⊢ (x{⟨A, y⟩}z ↔ ⟨x, z⟩ ∈ {⟨A, y⟩})
6		vex 2863	. . . . . . . . 9 ⊢ x ∈ V
7		vex 2863	. . . . . . . . 9 ⊢ z ∈ V
8	6, 7	opex 4589	. . . . . . . 8 ⊢ ⟨x, z⟩ ∈ V
9	8	elsnc 3757	. . . . . . 7 ⊢ (⟨x, z⟩ ∈ {⟨A, y⟩} ↔ ⟨x, z⟩ = ⟨A, y⟩)
10		opth 4603	. . . . . . . 8 ⊢ (⟨x, z⟩ = ⟨A, y⟩ ↔ (x = A ∧ z = y))
11		ancom 437	. . . . . . . 8 ⊢ ((x = A ∧ z = y) ↔ (z = y ∧ x = A))
12	10, 11	bitri 240	. . . . . . 7 ⊢ (⟨x, z⟩ = ⟨A, y⟩ ↔ (z = y ∧ x = A))
13	5, 9, 12	3bitri 262	. . . . . 6 ⊢ (x{⟨A, y⟩}z ↔ (z = y ∧ x = A))
14	13	exbii 1582	. . . . 5 ⊢ (∃z x{⟨A, y⟩}z ↔ ∃z(z = y ∧ x = A))
15		vex 2863	. . . . . 6 ⊢ y ∈ V
16		biidd 228	. . . . . 6 ⊢ (z = y → (x = A ↔ x = A))
17	15, 16	ceqsexv 2895	. . . . 5 ⊢ (∃z(z = y ∧ x = A) ↔ x = A)
18	14, 17	bitri 240	. . . 4 ⊢ (∃z x{⟨A, y⟩}z ↔ x = A)
19		eldm 4899	. . . 4 ⊢ (x ∈ dom {⟨A, y⟩} ↔ ∃z x{⟨A, y⟩}z)
20		elsn 3749	. . . 4 ⊢ (x ∈ {A} ↔ x = A)
21	18, 19, 20	3bitr4i 268	. . 3 ⊢ (x ∈ dom {⟨A, y⟩} ↔ x ∈ {A})
22	21	eqriv 2350	. 2 ⊢ dom {⟨A, y⟩} = {A}
23	4, 22	vtoclg 2915	1 ⊢ (B ∈ V → dom {⟨A, B⟩} = {A})

Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {csn 3738  ⟨cop 4562   class class class wbr 4640  dom cdm 4773

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-ima 4728  df-cnv 4786  df-rn 4787  df-dm 4788

This theorem is referenced by:  dmsnopss  5068  dmpropg  5069  dmsnop  5070  funprg  5150  funprgOLD  5151