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Theorem jech9.3 8850
 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 8803 . . 3 𝑅1 Fn On
2 fniunfv 6668 . . 3 (𝑅1 Fn On → 𝑥 ∈ On (𝑅1𝑥) = ran 𝑅1)
31, 2ax-mp 5 . 2 𝑥 ∈ On (𝑅1𝑥) = ran 𝑅1
4 fndm 6151 . . . . . 6 (𝑅1 Fn On → dom 𝑅1 = On)
51, 4ax-mp 5 . . . . 5 dom 𝑅1 = On
65imaeq2i 5622 . . . 4 (𝑅1 “ dom 𝑅1) = (𝑅1 “ On)
7 imadmrn 5634 . . . 4 (𝑅1 “ dom 𝑅1) = ran 𝑅1
86, 7eqtr3i 2784 . . 3 (𝑅1 “ On) = ran 𝑅1
98unieqi 4597 . 2 (𝑅1 “ On) = ran 𝑅1
10 unir1 8849 . 2 (𝑅1 “ On) = V
113, 9, 103eqtr2i 2788 1 𝑥 ∈ On (𝑅1𝑥) = V
 Colors of variables: wff setvar class Syntax hints:   = wceq 1632  Vcvv 3340  ∪ cuni 4588  ∪ ciun 4672  dom cdm 5266  ran crn 5267   “ cima 5269  Oncon0 5884   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  ax-reg 8662  ax-inf2 8711 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: (None)
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