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

Theorem r1elssi 8836
Description: The range of the 𝑅1 function is transitive. Lemma 2.10 of [Kunen] p. 97. One direction of r1elss 8837 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 4900 . . . 4 (∀𝑥 ∈ On Tr (𝑅1𝑥) → Tr 𝑥 ∈ On (𝑅1𝑥))
2 r1tr 8807 . . . . 5 Tr (𝑅1𝑥)
32a1i 11 . . . 4 (𝑥 ∈ On → Tr (𝑅1𝑥))
41, 3mprg 3075 . . 3 Tr 𝑥 ∈ On (𝑅1𝑥)
5 r1funlim 8797 . . . . . 6 (Fun 𝑅1 ∧ Lim dom 𝑅1)
65simpli 470 . . . . 5 Fun 𝑅1
7 funiunfv 6652 . . . . 5 (Fun 𝑅1 𝑥 ∈ On (𝑅1𝑥) = (𝑅1 “ On))
86, 7ax-mp 5 . . . 4 𝑥 ∈ On (𝑅1𝑥) = (𝑅1 “ On)
9 treq 4893 . . . 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 220 . 2 Tr (𝑅1 “ On)
12 trss 4896 . 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 196   = wceq 1631  wcel 2145  wss 3723   cuni 4575   ciun 4655  Tr wtr 4887  dom cdm 5250  cima 5253  Oncon0 5865  Lim wlim 5866  Fun wfun 6024  cfv 6030  𝑅1cr1 8793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-om 7217  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-r1 8795
This theorem is referenced by:  r1elss  8837  pwwf  8838  rankelb  8855  rankval3b  8857  r1pw  8876  rankuni2b  8884  tcwf  8914  tcrank  8915  hsmexlem4  9457  rankcf  9805  wfgru  9844  grur1  9848
  Copyright terms: Public domain W3C validator