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Theorem r1elss 8842
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 8841 . 2 (𝐴 (𝑅1 “ On) → 𝐴 (𝑅1 “ On))
2 r1elss.1 . . . 4 𝐴 ∈ V
32tz9.12 8826 . . 3 (∀𝑦𝐴𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥))
4 dfss3 3733 . . . 4 (𝐴 (𝑅1 “ On) ↔ ∀𝑦𝐴 𝑦 (𝑅1 “ On))
5 r1fnon 8803 . . . . . . . 8 𝑅1 Fn On
6 fnfun 6149 . . . . . . . 8 (𝑅1 Fn On → Fun 𝑅1)
7 funiunfv 6669 . . . . . . . 8 (Fun 𝑅1 𝑥 ∈ On (𝑅1𝑥) = (𝑅1 “ On))
85, 6, 7mp2b 10 . . . . . . 7 𝑥 ∈ On (𝑅1𝑥) = (𝑅1 “ On)
98eleq2i 2831 . . . . . 6 (𝑦 𝑥 ∈ On (𝑅1𝑥) ↔ 𝑦 (𝑅1 “ On))
10 eliun 4676 . . . . . 6 (𝑦 𝑥 ∈ On (𝑅1𝑥) ↔ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥))
119, 10bitr3i 266 . . . . 5 (𝑦 (𝑅1 “ On) ↔ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥))
1211ralbii 3118 . . . 4 (∀𝑦𝐴 𝑦 (𝑅1 “ On) ↔ ∀𝑦𝐴𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥))
134, 12bitri 264 . . 3 (𝐴 (𝑅1 “ On) ↔ ∀𝑦𝐴𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥))
148eleq2i 2831 . . . 4 (𝐴 𝑥 ∈ On (𝑅1𝑥) ↔ 𝐴 (𝑅1 “ On))
15 eliun 4676 . . . 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 1632  wcel 2139  wral 3050  wrex 3051  Vcvv 3340  wss 3715   cuni 4588   ciun 4672  cima 5269  Oncon0 5884  Fun wfun 6043   Fn wfn 6044  cfv 6049  𝑅1cr1 8798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-om 7231  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-r1 8800
This theorem is referenced by:  unir1  8849  tcwf  8919  tcrank  8920  rankcf  9791  wfgru  9830
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