Theorem rnr1 9811

Description: The range of the cumulative hierarchy of sets function. This is the class of all stages of the cumulative hierarchy of sets, not the class of all sets in some stage of the cumulative hierarchy (that is ∪ ran 𝑅₁).

This statement does not require the axiom of replacement by avoiding using r1fnon 9810. (Contributed by BJ, 27-Sep-2026.)

Assertion

Ref	Expression
rnr1	⊢ ran 𝑅1 = (𝑅1 “ On)

Proof of Theorem rnr1

Step	Hyp	Ref	Expression
1		r1dmlim 9809	. 2 ⊢ Lim dom 𝑅1
2		limord 6450	. . 3 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1)
3		ordsson 7831	. . 3 ⊢ (Ord dom 𝑅1 → dom 𝑅1 ⊆ On)
4		dfrn7 6084	. . 3 ⊢ (dom 𝑅1 ⊆ On → ran 𝑅1 = (𝑅1 “ On))
5	2, 3, 4	3syl 19	. 2 ⊢ (Lim dom 𝑅1 → ran 𝑅1 = (𝑅1 “ On))
6	1, 5	ax-mp 5	1 ⊢ ran 𝑅1 = (𝑅1 “ On)

This proof depends on syntax axioms:   = wceq 1570   ⊆ wss 3900  dom cdm 5658  ran crn 5659   “ cima 5661  Ord word 6387  Oncon0 6388  Lim wlim 6389  𝑅₁cr1 9803

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-r1 9805

This theorem is used by:  rankwflemb  9837