Theorem jech9.3 8677

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 8630	. . 3 ⊢ 𝑅1 Fn On
2		fniunfv 6505	. . 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 5990	. . . . . 6 ⊢ (𝑅1 Fn On → dom 𝑅1 = On)
5	1, 4	ax-mp 5	. . . . 5 ⊢ dom 𝑅1 = On
6	5	imaeq2i 5464	. . . 4 ⊢ (𝑅1 “ dom 𝑅1) = (𝑅1 “ On)
7		imadmrn 5476	. . . 4 ⊢ (𝑅1 “ dom 𝑅1) = ran 𝑅1
8	6, 7	eqtr3i 2646	. . 3 ⊢ (𝑅1 “ On) = ran 𝑅1
9	8	unieqi 4445	. 2 ⊢ ∪ (𝑅1 “ On) = ∪ ran 𝑅1
10		unir1 8676	. 2 ⊢ ∪ (𝑅1 “ On) = V
11	3, 9, 10	3eqtr2i 2650	1 ⊢ ∪ 𝑥 ∈ On (𝑅1‘𝑥) = V

Syntax hints:   = wceq 1483  Vcvv 3200  ∪ cuni 4436  ∪ ciun 4520  dom cdm 5114  ran crn 5115   “ cima 5117  Oncon0 5723   Fn wfn 5883  ‘cfv 5888  𝑅₁cr1 8625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-reg 8497  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-r1 8627

This theorem is referenced by: (None)