MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  r1elss Structured version   Visualization version   GIF version

Theorem r1elss 9797
Description: The range of the 𝑅1 function is transitive. Lemma 2.10 of [Kunen] p. 97. (Contributed by Mario Carneiro, 22-Mar-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)
Hypothesis
Ref Expression
r1elss.1 𝐴 ∈ V
Assertion
Ref Expression
r1elss (𝐴 (𝑅1 “ On) ↔ 𝐴 (𝑅1 “ On))

Proof of Theorem r1elss
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 r1elssi 9796 . 2 (𝐴 (𝑅1 “ On) → 𝐴 (𝑅1 “ On))
2 r1elss.1 . . . 4 𝐴 ∈ V
32tz9.12 9781 . . 3 (∀𝑦𝐴𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥))
4 dfss3 3969 . . . 4 (𝐴 (𝑅1 “ On) ↔ ∀𝑦𝐴 𝑦 (𝑅1 “ On))
5 r1fnon 9758 . . . . . . . 8 𝑅1 Fn On
6 fnfun 6646 . . . . . . . 8 (𝑅1 Fn On → Fun 𝑅1)
7 funiunfv 7242 . . . . . . . 8 (Fun 𝑅1 𝑥 ∈ On (𝑅1𝑥) = (𝑅1 “ On))
85, 6, 7mp2b 10 . . . . . . 7 𝑥 ∈ On (𝑅1𝑥) = (𝑅1 “ On)
98eleq2i 2826 . . . . . 6 (𝑦 𝑥 ∈ On (𝑅1𝑥) ↔ 𝑦 (𝑅1 “ On))
10 eliun 5000 . . . . . 6 (𝑦 𝑥 ∈ On (𝑅1𝑥) ↔ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥))
119, 10bitr3i 277 . . . . 5 (𝑦 (𝑅1 “ On) ↔ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥))
1211ralbii 3094 . . . 4 (∀𝑦𝐴 𝑦 (𝑅1 “ On) ↔ ∀𝑦𝐴𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥))
134, 12bitri 275 . . 3 (𝐴 (𝑅1 “ On) ↔ ∀𝑦𝐴𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥))
148eleq2i 2826 . . . 4 (𝐴 𝑥 ∈ On (𝑅1𝑥) ↔ 𝐴 (𝑅1 “ On))
15 eliun 5000 . . . 4 (𝐴 𝑥 ∈ On (𝑅1𝑥) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥))
1614, 15bitr3i 277 . . 3 (𝐴 (𝑅1 “ On) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥))
173, 13, 163imtr4i 292 . 2 (𝐴 (𝑅1 “ On) → 𝐴 (𝑅1 “ On))
181, 17impbii 208 1 (𝐴 (𝑅1 “ On) ↔ 𝐴 (𝑅1 “ On))
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1542  wcel 2107  wral 3062  wrex 3071  Vcvv 3475  wss 3947   cuni 4907   ciun 4996  cima 5678  Oncon0 6361  Fun wfun 6534   Fn wfn 6535  cfv 6540  𝑅1cr1 9753
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7407  df-om 7851  df-2nd 7971  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-r1 9755
This theorem is referenced by:  unir1  9804  tcwf  9874  tcrank  9875  rankcf  10768  wfgru  10807
  Copyright terms: Public domain W3C validator