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Theorem r1elss 9846
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 9845 . 2 (𝐴 (𝑅1 “ On) → 𝐴 (𝑅1 “ On))
2 r1elss.1 . . . 4 𝐴 ∈ V
32tz9.12 9830 . . 3 (∀𝑦𝐴𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥))
4 dfss3 3972 . . . 4 (𝐴 (𝑅1 “ On) ↔ ∀𝑦𝐴 𝑦 (𝑅1 “ On))
5 r1fnon 9807 . . . . . . . 8 𝑅1 Fn On
6 fnfun 6668 . . . . . . . 8 (𝑅1 Fn On → Fun 𝑅1)
7 funiunfv 7268 . . . . . . . 8 (Fun 𝑅1 𝑥 ∈ On (𝑅1𝑥) = (𝑅1 “ On))
85, 6, 7mp2b 10 . . . . . . 7 𝑥 ∈ On (𝑅1𝑥) = (𝑅1 “ On)
98eleq2i 2833 . . . . . 6 (𝑦 𝑥 ∈ On (𝑅1𝑥) ↔ 𝑦 (𝑅1 “ On))
10 eliun 4995 . . . . . 6 (𝑦 𝑥 ∈ On (𝑅1𝑥) ↔ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥))
119, 10bitr3i 277 . . . . 5 (𝑦 (𝑅1 “ On) ↔ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥))
1211ralbii 3093 . . . 4 (∀𝑦𝐴 𝑦 (𝑅1 “ On) ↔ ∀𝑦𝐴𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥))
134, 12bitri 275 . . 3 (𝐴 (𝑅1 “ On) ↔ ∀𝑦𝐴𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥))
148eleq2i 2833 . . . 4 (𝐴 𝑥 ∈ On (𝑅1𝑥) ↔ 𝐴 (𝑅1 “ On))
15 eliun 4995 . . . 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 209 1 (𝐴 (𝑅1 “ On) ↔ 𝐴 (𝑅1 “ On))
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1540  wcel 2108  wral 3061  wrex 3070  Vcvv 3480  wss 3951   cuni 4907   ciun 4991  cima 5688  Oncon0 6384  Fun wfun 6555   Fn wfn 6556  cfv 6561  𝑅1cr1 9802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-r1 9804
This theorem is referenced by:  unir1  9853  tcwf  9923  tcrank  9924  rankcf  10817  wfgru  10856  trfr  44979  tcfr  44980  wfaxrep  45011
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