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Theorem clos1basesucg 5884
 Description: A member of a closure is either in the base set or connected to another member by R. Theorem IX.5.16 of [Rosser] p. 248. (Contributed by Scott Fenton, 31-Jul-2019.)
Hypothesis
Ref Expression
clos1basesucg.1 C = Clos1 (S, R)
Assertion
Ref Expression
clos1basesucg ((S V R W) → (A C ↔ (A S x C xRA)))
Distinct variable groups:   x,A   x,S   x,R   x,C
Allowed substitution hints:   V(x)   W(x)

Proof of Theorem clos1basesucg
Dummy variables s r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 clos1eq1 5874 . . . . 5 (s = S Clos1 (s, r) = Clos1 (S, r))
21eleq2d 2420 . . . 4 (s = S → (A Clos1 (s, r) ↔ A Clos1 (S, r)))
3 eleq2 2414 . . . . 5 (s = S → (A sA S))
41rexeqdv 2814 . . . . 5 (s = S → (x Clos1 (s, r)xrAx Clos1 (S, r)xrA))
53, 4orbi12d 690 . . . 4 (s = S → ((A s x Clos1 (s, r)xrA) ↔ (A S x Clos1 (S, r)xrA)))
62, 5bibi12d 312 . . 3 (s = S → ((A Clos1 (s, r) ↔ (A s x Clos1 (s, r)xrA)) ↔ (A Clos1 (S, r) ↔ (A S x Clos1 (S, r)xrA))))
7 clos1eq2 5875 . . . . 5 (r = R Clos1 (S, r) = Clos1 (S, R))
87eleq2d 2420 . . . 4 (r = R → (A Clos1 (S, r) ↔ A Clos1 (S, R)))
9 breq 4641 . . . . . 6 (r = R → (xrAxRA))
107, 9rexeqbidv 2820 . . . . 5 (r = R → (x Clos1 (S, r)xrAx Clos1 (S, R)xRA))
1110orbi2d 682 . . . 4 (r = R → ((A S x Clos1 (S, r)xrA) ↔ (A S x Clos1 (S, R)xRA)))
128, 11bibi12d 312 . . 3 (r = R → ((A Clos1 (S, r) ↔ (A S x Clos1 (S, r)xrA)) ↔ (A Clos1 (S, R) ↔ (A S x Clos1 (S, R)xRA))))
13 vex 2862 . . . 4 s V
14 vex 2862 . . . 4 r V
15 eqid 2353 . . . 4 Clos1 (s, r) = Clos1 (s, r)
1613, 14, 15clos1basesuc 5882 . . 3 (A Clos1 (s, r) ↔ (A s x Clos1 (s, r)xrA))
176, 12, 16vtocl2g 2918 . 2 ((S V R W) → (A Clos1 (S, R) ↔ (A S x Clos1 (S, R)xRA)))
18 clos1basesucg.1 . . 3 C = Clos1 (S, R)
1918eleq2i 2417 . 2 (A CA Clos1 (S, R))
2018rexeqi 2812 . . 3 (x C xRAx Clos1 (S, R)xRA)
2120orbi2i 505 . 2 ((A S x C xRA) ↔ (A S x Clos1 (S, R)xRA))
2217, 19, 213bitr4g 279 1 ((S V R W) → (A C ↔ (A S x C xRA)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃wrex 2615   class class class wbr 4639   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:  dmfrec  6316  fnfreclem2  6318  fnfreclem3  6319  frec0  6321  frecsuc  6322
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