Theorem rnr1 9774

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 9773. (Contributed by BJ, 27-Sep-2026.)

Assertion

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

Proof of Theorem rnr1

Step	Hyp	Ref	Expression
1		r1dmlim 9772	. 2 ⊢ Lim dom 𝑅1
2		limord 6424	. . 3 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1)
3		ordsson 7797	. . 3 ⊢ (Ord dom 𝑅1 → dom 𝑅1 ⊆ On)
4		dfrn7 6065	. . 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 1569   ⊆ wss 3895  dom cdm 5650  ran crn 5651   “ cima 5653  Ord word 6361  Oncon0 6362  Lim wlim 6363  𝑅₁cr1 9766

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-r1 9768

This theorem is used by:  rankwflemb  9800