Theorem dmsi 5520

Description: Calculate the domain of a singleton image. Theorem X.4.29.I of [Rosser] p. 301. (Contributed by SF, 26-Feb-2015.)

Assertion

Ref	Expression
dmsi	⊢ dom SI R = ℘1dom R

Proof of Theorem dmsi

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

Step	Hyp	Ref	Expression
1		3anass 938	. . . . . . . 8 ⊢ ((x = {a} ∧ y = {b} ∧ aRb) ↔ (x = {a} ∧ (y = {b} ∧ aRb)))
2	1	2exbii 1583	. . . . . . 7 ⊢ (∃y∃b(x = {a} ∧ y = {b} ∧ aRb) ↔ ∃y∃b(x = {a} ∧ (y = {b} ∧ aRb)))
3		19.42vv 1907	. . . . . . 7 ⊢ (∃y∃b(x = {a} ∧ (y = {b} ∧ aRb)) ↔ (x = {a} ∧ ∃y∃b(y = {b} ∧ aRb)))
4	2, 3	bitri 240	. . . . . 6 ⊢ (∃y∃b(x = {a} ∧ y = {b} ∧ aRb) ↔ (x = {a} ∧ ∃y∃b(y = {b} ∧ aRb)))
5		snex 4112	. . . . . . . . . . . 12 ⊢ {b} ∈ V
6	5	isseti 2866	. . . . . . . . . . 11 ⊢ ∃y y = {b}
7		19.41v 1901	. . . . . . . . . . 11 ⊢ (∃y(y = {b} ∧ aRb) ↔ (∃y y = {b} ∧ aRb))
8	6, 7	mpbiran 884	. . . . . . . . . 10 ⊢ (∃y(y = {b} ∧ aRb) ↔ aRb)
9	8	exbii 1582	. . . . . . . . 9 ⊢ (∃b∃y(y = {b} ∧ aRb) ↔ ∃b aRb)
10		excom 1741	. . . . . . . . 9 ⊢ (∃y∃b(y = {b} ∧ aRb) ↔ ∃b∃y(y = {b} ∧ aRb))
11		eldm 4899	. . . . . . . . 9 ⊢ (a ∈ dom R ↔ ∃b aRb)
12	9, 10, 11	3bitr4i 268	. . . . . . . 8 ⊢ (∃y∃b(y = {b} ∧ aRb) ↔ a ∈ dom R)
13	12	anbi2i 675	. . . . . . 7 ⊢ ((x = {a} ∧ ∃y∃b(y = {b} ∧ aRb)) ↔ (x = {a} ∧ a ∈ dom R))
14		ancom 437	. . . . . . 7 ⊢ ((x = {a} ∧ a ∈ dom R) ↔ (a ∈ dom R ∧ x = {a}))
15	13, 14	bitri 240	. . . . . 6 ⊢ ((x = {a} ∧ ∃y∃b(y = {b} ∧ aRb)) ↔ (a ∈ dom R ∧ x = {a}))
16	4, 15	bitri 240	. . . . 5 ⊢ (∃y∃b(x = {a} ∧ y = {b} ∧ aRb) ↔ (a ∈ dom R ∧ x = {a}))
17	16	exbii 1582	. . . 4 ⊢ (∃a∃y∃b(x = {a} ∧ y = {b} ∧ aRb) ↔ ∃a(a ∈ dom R ∧ x = {a}))
18		excom 1741	. . . 4 ⊢ (∃y∃a∃b(x = {a} ∧ y = {b} ∧ aRb) ↔ ∃a∃y∃b(x = {a} ∧ y = {b} ∧ aRb))
19		df-rex 2621	. . . 4 ⊢ (∃a ∈ dom R x = {a} ↔ ∃a(a ∈ dom R ∧ x = {a}))
20	17, 18, 19	3bitr4i 268	. . 3 ⊢ (∃y∃a∃b(x = {a} ∧ y = {b} ∧ aRb) ↔ ∃a ∈ dom R x = {a})
21		eldm 4899	. . . 4 ⊢ (x ∈ dom SI R ↔ ∃y x SI Ry)
22		brsi 4762	. . . . 5 ⊢ (x SI Ry ↔ ∃a∃b(x = {a} ∧ y = {b} ∧ aRb))
23	22	exbii 1582	. . . 4 ⊢ (∃y x SI Ry ↔ ∃y∃a∃b(x = {a} ∧ y = {b} ∧ aRb))
24	21, 23	bitri 240	. . 3 ⊢ (x ∈ dom SI R ↔ ∃y∃a∃b(x = {a} ∧ y = {b} ∧ aRb))
25		elpw1 4145	. . 3 ⊢ (x ∈ ℘1dom R ↔ ∃a ∈ dom R x = {a})
26	20, 24, 25	3bitr4i 268	. 2 ⊢ (x ∈ dom SI R ↔ x ∈ ℘1dom R)
27	26	eqriv 2350	1 ⊢ dom SI R = ℘1dom R

Syntax hints:   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2616  {csn 3738  ℘₁cpw1 4136   class class class wbr 4640   SI csi 4721  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-si 4729  df-cnv 4786  df-rn 4787  df-dm 4788

This theorem is referenced by:  rnsi  5522  enpw1  6063