New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  dmsi GIF version

Theorem dmsi 5519
 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.
StepHypRef Expression
1 3anass 938 . . . . . . . 8 ((x = {a} y = {b} aRb) ↔ (x = {a} (y = {b} aRb)))
212exbii 1583 . . . . . . 7 (yb(x = {a} y = {b} aRb) ↔ yb(x = {a} (y = {b} aRb)))
3 19.42vv 1907 . . . . . . 7 (yb(x = {a} (y = {b} aRb)) ↔ (x = {a} yb(y = {b} aRb)))
42, 3bitri 240 . . . . . 6 (yb(x = {a} y = {b} aRb) ↔ (x = {a} yb(y = {b} aRb)))
5 snex 4111 . . . . . . . . . . . 12 {b} V
65isseti 2865 . . . . . . . . . . 11 y y = {b}
7 19.41v 1901 . . . . . . . . . . 11 (y(y = {b} aRb) ↔ (y y = {b} aRb))
86, 7mpbiran 884 . . . . . . . . . 10 (y(y = {b} aRb) ↔ aRb)
98exbii 1582 . . . . . . . . 9 (by(y = {b} aRb) ↔ b aRb)
10 excom 1741 . . . . . . . . 9 (yb(y = {b} aRb) ↔ by(y = {b} aRb))
11 eldm 4898 . . . . . . . . 9 (a dom Rb aRb)
129, 10, 113bitr4i 268 . . . . . . . 8 (yb(y = {b} aRb) ↔ a dom R)
1312anbi2i 675 . . . . . . 7 ((x = {a} yb(y = {b} aRb)) ↔ (x = {a} a dom R))
14 ancom 437 . . . . . . 7 ((x = {a} a dom R) ↔ (a dom R x = {a}))
1513, 14bitri 240 . . . . . 6 ((x = {a} yb(y = {b} aRb)) ↔ (a dom R x = {a}))
164, 15bitri 240 . . . . 5 (yb(x = {a} y = {b} aRb) ↔ (a dom R x = {a}))
1716exbii 1582 . . . 4 (ayb(x = {a} y = {b} aRb) ↔ a(a dom R x = {a}))
18 excom 1741 . . . 4 (yab(x = {a} y = {b} aRb) ↔ ayb(x = {a} y = {b} aRb))
19 df-rex 2620 . . . 4 (a dom R x = {a} ↔ a(a dom R x = {a}))
2017, 18, 193bitr4i 268 . . 3 (yab(x = {a} y = {b} aRb) ↔ a dom R x = {a})
21 eldm 4898 . . . 4 (x dom SI Ry x SI Ry)
22 brsi 4761 . . . . 5 (x SI Ryab(x = {a} y = {b} aRb))
2322exbii 1582 . . . 4 (y x SI Ryyab(x = {a} y = {b} aRb))
2421, 23bitri 240 . . 3 (x dom SI Ryab(x = {a} y = {b} aRb))
25 elpw1 4144 . . 3 (x 1dom Ra dom R x = {a})
2620, 24, 253bitr4i 268 . 2 (x dom SI Rx 1dom R)
2726eqriv 2350 1 dom SI R = 1dom R
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  {csn 3737  ℘1cpw1 4135   class class class wbr 4639   SI csi 4720  dom cdm 4772 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-ima 4727  df-si 4728  df-cnv 4785  df-rn 4786  df-dm 4787 This theorem is referenced by:  rnsi  5521  enpw1  6062
 Copyright terms: Public domain W3C validator