MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  r1elssi Structured version   Visualization version   GIF version

Theorem r1elssi 9210
Description: The range of the 𝑅1 function is transitive. Lemma 2.10 of [Kunen] p. 97. One direction of r1elss 9211 that doesn't need 𝐴 to be a set. (Contributed by Mario Carneiro, 22-Mar-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
r1elssi (𝐴 (𝑅1 “ On) → 𝐴 (𝑅1 “ On))

Proof of Theorem r1elssi
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 triun 5158 . . . 4 (∀𝑥 ∈ On Tr (𝑅1𝑥) → Tr 𝑥 ∈ On (𝑅1𝑥))
2 r1tr 9181 . . . . 5 Tr (𝑅1𝑥)
32a1i 11 . . . 4 (𝑥 ∈ On → Tr (𝑅1𝑥))
41, 3mprg 3140 . . 3 Tr 𝑥 ∈ On (𝑅1𝑥)
5 r1funlim 9171 . . . . . 6 (Fun 𝑅1 ∧ Lim dom 𝑅1)
65simpli 487 . . . . 5 Fun 𝑅1
7 funiunfv 6981 . . . . 5 (Fun 𝑅1 𝑥 ∈ On (𝑅1𝑥) = (𝑅1 “ On))
86, 7ax-mp 5 . . . 4 𝑥 ∈ On (𝑅1𝑥) = (𝑅1 “ On)
9 treq 5151 . . . 4 ( 𝑥 ∈ On (𝑅1𝑥) = (𝑅1 “ On) → (Tr 𝑥 ∈ On (𝑅1𝑥) ↔ Tr (𝑅1 “ On)))
108, 9ax-mp 5 . . 3 (Tr 𝑥 ∈ On (𝑅1𝑥) ↔ Tr (𝑅1 “ On))
114, 10mpbi 233 . 2 Tr (𝑅1 “ On)
12 trss 5154 . 2 (Tr (𝑅1 “ On) → (𝐴 (𝑅1 “ On) → 𝐴 (𝑅1 “ On)))
1311, 12ax-mp 5 1 (𝐴 (𝑅1 “ On) → 𝐴 (𝑅1 “ On))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wcel 2115  wss 3910   cuni 4811   ciun 4892  Tr wtr 5145  dom cdm 5528  cima 5531  Oncon0 6164  Lim wlim 6165  Fun wfun 6322  cfv 6328  𝑅1cr1 9167
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-om 7556  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-r1 9169
This theorem is referenced by:  r1elss  9211  pwwf  9212  rankelb  9229  rankval3b  9231  r1pw  9250  rankuni2b  9258  tcwf  9288  tcrank  9289  hsmexlem4  9828  rankcf  10176  wfgru  10215  grur1  10219
  Copyright terms: Public domain W3C validator