Theorem jech9.3 9729

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 (𝑅1‘𝑥) = V

Proof of Theorem jech9.3

Step	Hyp	Ref	Expression
1		r1fnon 9682	. . 3 ⊢ 𝑅1 Fn On
2		fniunfv 7191	. . 3 ⊢ (𝑅1 Fn On → ∪ 𝑥 ∈ On (𝑅1‘𝑥) = ∪ ran 𝑅1)
3	1, 2	ax-mp 5	. 2 ⊢ ∪ 𝑥 ∈ On (𝑅1‘𝑥) = ∪ ran 𝑅1
4		fndm 6588	. . . . . 6 ⊢ (𝑅1 Fn On → dom 𝑅1 = On)
5	1, 4	ax-mp 5	. . . . 5 ⊢ dom 𝑅1 = On
6	5	imaeq2i 6010	. . . 4 ⊢ (𝑅1 “ dom 𝑅1) = (𝑅1 “ On)
7		imadmrn 6022	. . . 4 ⊢ (𝑅1 “ dom 𝑅1) = ran 𝑅1
8	6, 7	eqtr3i 2764	. . 3 ⊢ (𝑅1 “ On) = ran 𝑅1
9	8	unieqi 4850	. 2 ⊢ ∪ (𝑅1 “ On) = ∪ ran 𝑅1
10		unir1 9728	. 2 ⊢ ∪ (𝑅1 “ On) = V
11	3, 9, 10	3eqtr2i 2768	1 ⊢ ∪ 𝑥 ∈ On (𝑅1‘𝑥) = V

Syntax hints:   = wceq 1547  Vcvv 3431  ∪ cuni 4838  ∪ ciun 4921  dom cdm 5618  ran crn 5619   “ cima 5621  Oncon0 6310   Fn wfn 6480  ‘cfv 6485  𝑅₁cr1 9677

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-reg 9497  ax-inf2 9553

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-r1 9679

This theorem is referenced by: (None)