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Theorem r1elss 8707
 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 8706 . 2 (𝐴 (𝑅1 “ On) → 𝐴 (𝑅1 “ On))
2 r1elss.1 . . . 4 𝐴 ∈ V
32tz9.12 8691 . . 3 (∀𝑦𝐴𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥))
4 dfss3 3625 . . . 4 (𝐴 (𝑅1 “ On) ↔ ∀𝑦𝐴 𝑦 (𝑅1 “ On))
5 r1fnon 8668 . . . . . . . 8 𝑅1 Fn On
6 fnfun 6026 . . . . . . . 8 (𝑅1 Fn On → Fun 𝑅1)
7 funiunfv 6546 . . . . . . . 8 (Fun 𝑅1 𝑥 ∈ On (𝑅1𝑥) = (𝑅1 “ On))
85, 6, 7mp2b 10 . . . . . . 7 𝑥 ∈ On (𝑅1𝑥) = (𝑅1 “ On)
98eleq2i 2722 . . . . . 6 (𝑦 𝑥 ∈ On (𝑅1𝑥) ↔ 𝑦 (𝑅1 “ On))
10 eliun 4556 . . . . . 6 (𝑦 𝑥 ∈ On (𝑅1𝑥) ↔ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥))
119, 10bitr3i 266 . . . . 5 (𝑦 (𝑅1 “ On) ↔ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥))
1211ralbii 3009 . . . 4 (∀𝑦𝐴 𝑦 (𝑅1 “ On) ↔ ∀𝑦𝐴𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥))
134, 12bitri 264 . . 3 (𝐴 (𝑅1 “ On) ↔ ∀𝑦𝐴𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥))
148eleq2i 2722 . . . 4 (𝐴 𝑥 ∈ On (𝑅1𝑥) ↔ 𝐴 (𝑅1 “ On))
15 eliun 4556 . . . 4 (𝐴 𝑥 ∈ On (𝑅1𝑥) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥))
1614, 15bitr3i 266 . . 3 (𝐴 (𝑅1 “ On) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥))
173, 13, 163imtr4i 281 . 2 (𝐴 (𝑅1 “ On) → 𝐴 (𝑅1 “ On))
181, 17impbii 199 1 (𝐴 (𝑅1 “ On) ↔ 𝐴 (𝑅1 “ On))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   = wceq 1523   ∈ wcel 2030  ∀wral 2941  ∃wrex 2942  Vcvv 3231   ⊆ wss 3607  ∪ cuni 4468  ∪ ciun 4552   “ cima 5146  Oncon0 5761  Fun wfun 5920   Fn wfn 5921  ‘cfv 5926  𝑅1cr1 8663 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-r1 8665 This theorem is referenced by:  unir1  8714  tcwf  8784  tcrank  8785  rankcf  9637  wfgru  9676
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