Theorem rnwf 45658

Description: The range of a well-founded set is well-founded. (Contributed by Eric Schmidt, 12-Sep-2025.)

Assertion

Ref	Expression
rnwf	⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ran 𝐴 ∈ ∪ (𝑅1 “ On))

Proof of Theorem rnwf

Step	Hyp	Ref	Expression
1		uniwf 9792	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ ∪ 𝐴 ∈ ∪ (𝑅1 “ On))
2		uniwf 9792	. . 3 ⊢ (∪ 𝐴 ∈ ∪ (𝑅1 “ On) ↔ ∪ ∪ 𝐴 ∈ ∪ (𝑅1 “ On))
3	1, 2	bitri 278	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ ∪ ∪ 𝐴 ∈ ∪ (𝑅1 “ On))
4		ssun2 4133	. . . 4 ⊢ ran 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)
5		dmrnssfld 5966	. . . 4 ⊢ (dom 𝐴 ∪ ran 𝐴) ⊆ ∪ ∪ 𝐴
6	4, 5	sstri 3947	. . 3 ⊢ ran 𝐴 ⊆ ∪ ∪ 𝐴
7		sswf 9781	. . 3 ⊢ ((∪ ∪ 𝐴 ∈ ∪ (𝑅1 “ On) ∧ ran 𝐴 ⊆ ∪ ∪ 𝐴) → ran 𝐴 ∈ ∪ (𝑅1 “ On))
8	6, 7	mpan2 703	. 2 ⊢ (∪ ∪ 𝐴 ∈ ∪ (𝑅1 “ On) → ran 𝐴 ∈ ∪ (𝑅1 “ On))
9	3, 8	sylbi 220	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ran 𝐴 ∈ ∪ (𝑅1 “ On))

Syntax hints:   → wi 4   ∈ wcel 2143   ∪ cun 3904   ⊆ wss 3906  ∪ cuni 4873  dom cdm 5663  ran crn 5664   “ cima 5666  Oncon0 6362  𝑅₁cr1 9735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  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 7415  df-om 7864  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-r1 9737  df-rank 9738

This theorem is referenced by:  relwf  45659