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Theorem jech9.3 9815
Description: Every set belongs to some value of the cumulative hierarchy of sets function 𝑅1, i.e. the indexed union of all values of 𝑅1 is the universe. Lemma 9.3 of [Jech] p. 71. (Contributed by NM, 4-Oct-2003.) (Revised by Mario Carneiro, 8-Jun-2013.)
Assertion
Ref Expression
jech9.3 𝑥 ∈ On (𝑅1𝑥) = V

Proof of Theorem jech9.3
StepHypRef Expression
1 r1fnon 9768 . . 3 𝑅1 Fn On
2 fniunfv 7249 . . 3 (𝑅1 Fn On → 𝑥 ∈ On (𝑅1𝑥) = ran 𝑅1)
31, 2ax-mp 5 . 2 𝑥 ∈ On (𝑅1𝑥) = ran 𝑅1
4 fndm 6652 . . . . . 6 (𝑅1 Fn On → dom 𝑅1 = On)
51, 4ax-mp 5 . . . . 5 dom 𝑅1 = On
65imaeq2i 6057 . . . 4 (𝑅1 “ dom 𝑅1) = (𝑅1 “ On)
7 imadmrn 6069 . . . 4 (𝑅1 “ dom 𝑅1) = ran 𝑅1
86, 7eqtr3i 2761 . . 3 (𝑅1 “ On) = ran 𝑅1
98unieqi 4921 . 2 (𝑅1 “ On) = ran 𝑅1
10 unir1 9814 . 2 (𝑅1 “ On) = V
113, 9, 103eqtr2i 2765 1 𝑥 ∈ On (𝑅1𝑥) = V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  Vcvv 3473   cuni 4908   ciun 4997  dom cdm 5676  ran crn 5677  cima 5679  Oncon0 6364   Fn wfn 6538  cfv 6543  𝑅1cr1 9763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-reg 9593  ax-inf2 9642
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  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 7415  df-om 7860  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-r1 9765
This theorem is referenced by: (None)
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