Theorem rnr1 9767

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 9766. (Contributed by BJ, 27-Sep-2026.)

Assertion

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

Proof of Theorem rnr1

Step	Hyp	Ref	Expression
1		r1dmlim 9765	. 2 ⊢ Lim dom 𝑅1
2		limord 6423	. . 3 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1)
3		ordsson 7795	. . 3 ⊢ (Ord dom 𝑅1 → dom 𝑅1 ⊆ On)
4		dfrn7 6066	. . 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 3899  dom cdm 5651  ran crn 5652   “ cima 5654  Ord word 6360  Oncon0 6361  Lim wlim 6362  𝑅₁cr1 9759

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7749

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7421  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-r1 9761

This theorem is used by:  rankwflemb  9793