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Theorem r1elss 9800
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 9799 . 2 (𝐴 (𝑅1 “ On) → 𝐴 (𝑅1 “ On))
2 r1elss.1 . . . 4 𝐴 ∈ V
32tz9.12 9784 . . 3 (∀𝑦𝐴𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥))
4 dfss3 3970 . . . 4 (𝐴 (𝑅1 “ On) ↔ ∀𝑦𝐴 𝑦 (𝑅1 “ On))
5 r1fnon 9761 . . . . . . . 8 𝑅1 Fn On
6 fnfun 6649 . . . . . . . 8 (𝑅1 Fn On → Fun 𝑅1)
7 funiunfv 7246 . . . . . . . 8 (Fun 𝑅1 𝑥 ∈ On (𝑅1𝑥) = (𝑅1 “ On))
85, 6, 7mp2b 10 . . . . . . 7 𝑥 ∈ On (𝑅1𝑥) = (𝑅1 “ On)
98eleq2i 2825 . . . . . 6 (𝑦 𝑥 ∈ On (𝑅1𝑥) ↔ 𝑦 (𝑅1 “ On))
10 eliun 5001 . . . . . 6 (𝑦 𝑥 ∈ On (𝑅1𝑥) ↔ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥))
119, 10bitr3i 276 . . . . 5 (𝑦 (𝑅1 “ On) ↔ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥))
1211ralbii 3093 . . . 4 (∀𝑦𝐴 𝑦 (𝑅1 “ On) ↔ ∀𝑦𝐴𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥))
134, 12bitri 274 . . 3 (𝐴 (𝑅1 “ On) ↔ ∀𝑦𝐴𝑥 ∈ On 𝑦 ∈ (𝑅1𝑥))
148eleq2i 2825 . . . 4 (𝐴 𝑥 ∈ On (𝑅1𝑥) ↔ 𝐴 (𝑅1 “ On))
15 eliun 5001 . . . 4 (𝐴 𝑥 ∈ On (𝑅1𝑥) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥))
1614, 15bitr3i 276 . . 3 (𝐴 (𝑅1 “ On) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥))
173, 13, 163imtr4i 291 . 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 1541  wcel 2106  wral 3061  wrex 3070  Vcvv 3474  wss 3948   cuni 4908   ciun 4997  cima 5679  Oncon0 6364  Fun wfun 6537   Fn wfn 6538  cfv 6543  𝑅1cr1 9756
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-om 7855  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-r1 9758
This theorem is referenced by:  unir1  9807  tcwf  9877  tcrank  9878  rankcf  10771  wfgru  10810
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