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Theorem jech9.3 9231
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 9184 . . 3 𝑅1 Fn On
2 fniunfv 6997 . . 3 (𝑅1 Fn On → 𝑥 ∈ On (𝑅1𝑥) = ran 𝑅1)
31, 2ax-mp 5 . 2 𝑥 ∈ On (𝑅1𝑥) = ran 𝑅1
4 fndm 6448 . . . . . 6 (𝑅1 Fn On → dom 𝑅1 = On)
51, 4ax-mp 5 . . . . 5 dom 𝑅1 = On
65imaeq2i 5920 . . . 4 (𝑅1 “ dom 𝑅1) = (𝑅1 “ On)
7 imadmrn 5932 . . . 4 (𝑅1 “ dom 𝑅1) = ran 𝑅1
86, 7eqtr3i 2843 . . 3 (𝑅1 “ On) = ran 𝑅1
98unieqi 4839 . 2 (𝑅1 “ On) = ran 𝑅1
10 unir1 9230 . 2 (𝑅1 “ On) = V
113, 9, 103eqtr2i 2847 1 𝑥 ∈ On (𝑅1𝑥) = V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  Vcvv 3492   cuni 4830   ciun 4910  dom cdm 5548  ran crn 5549  cima 5551  Oncon0 6184   Fn wfn 6343  cfv 6348  𝑅1cr1 9179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-reg 9044  ax-inf2 9092
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-om 7570  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-r1 9181
This theorem is referenced by: (None)
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