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Theorem clos1baseima 5883
 Description: A closure is equal to the base set together with the image of the closure under R. Theorem X.4.37 of [Rosser] p. 303. (Contributed by SF, 10-Mar-2015.)
Hypotheses
Ref Expression
clos1basesuc.1 S V
clos1basesuc.2 R V
clos1basesuc.3 C = Clos1 (S, R)
Assertion
Ref Expression
clos1baseima C = (S ∪ (RC))

Proof of Theorem clos1baseima
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elima 4754 . . . 4 (x (RC) ↔ y C yRx)
21orbi2i 505 . . 3 ((x S x (RC)) ↔ (x S y C yRx))
3 elun 3220 . . 3 (x (S ∪ (RC)) ↔ (x S x (RC)))
4 clos1basesuc.1 . . . 4 S V
5 clos1basesuc.2 . . . 4 R V
6 clos1basesuc.3 . . . 4 C = Clos1 (S, R)
74, 5, 6clos1basesuc 5882 . . 3 (x C ↔ (x S y C yRx))
82, 3, 73bitr4ri 269 . 2 (x Cx (S ∪ (RC)))
98eqriv 2350 1 C = (S ∪ (RC))
 Colors of variables: wff setvar class Syntax hints:   ∨ wo 357   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  Vcvv 2859   ∪ cun 3207   class class class wbr 4639   “ cima 4722   Clos1 cclos1 5872 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-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-2nd 4797  df-txp 5736  df-fix 5740  df-ins2 5750  df-ins3 5752  df-image 5754  df-clos1 5873 This theorem is referenced by:  sbthlem1  6203  spacssnc  6284
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