Theorem relwf 45656

Description: A relation is a well-founded set iff its domain and range are. (Contributed by Eric Schmidt, 29-Sep-2025.)

Assertion

Ref	Expression
relwf	⊢ (Rel 𝑅 → (𝑅 ∈ ∪ (𝑅1 “ On) ↔ (dom 𝑅 ∈ ∪ (𝑅1 “ On) ∧ ran 𝑅 ∈ ∪ (𝑅1 “ On))))

Proof of Theorem relwf

Step	Hyp	Ref	Expression
1		dmwf 45654	. . 3 ⊢ (𝑅 ∈ ∪ (𝑅1 “ On) → dom 𝑅 ∈ ∪ (𝑅1 “ On))
2		rnwf 45655	. . 3 ⊢ (𝑅 ∈ ∪ (𝑅1 “ On) → ran 𝑅 ∈ ∪ (𝑅1 “ On))
3	1, 2	jca 520	. 2 ⊢ (𝑅 ∈ ∪ (𝑅1 “ On) → (dom 𝑅 ∈ ∪ (𝑅1 “ On) ∧ ran 𝑅 ∈ ∪ (𝑅1 “ On)))
4		xpwf 45653	. . 3 ⊢ ((dom 𝑅 ∈ ∪ (𝑅1 “ On) ∧ ran 𝑅 ∈ ∪ (𝑅1 “ On)) → (dom 𝑅 × ran 𝑅) ∈ ∪ (𝑅1 “ On))
5		relssdmrn 6272	. . . . 5 ⊢ (Rel 𝑅 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅))
6		sswf 9781	. . . . 5 ⊢ (((dom 𝑅 × ran 𝑅) ∈ ∪ (𝑅1 “ On) ∧ 𝑅 ⊆ (dom 𝑅 × ran 𝑅)) → 𝑅 ∈ ∪ (𝑅1 “ On))
7	5, 6	sylan2 604	. . . 4 ⊢ (((dom 𝑅 × ran 𝑅) ∈ ∪ (𝑅1 “ On) ∧ Rel 𝑅) → 𝑅 ∈ ∪ (𝑅1 “ On))
8	7	expcom 418	. . 3 ⊢ (Rel 𝑅 → ((dom 𝑅 × ran 𝑅) ∈ ∪ (𝑅1 “ On) → 𝑅 ∈ ∪ (𝑅1 “ On)))
9	4, 8	syl5 35	. 2 ⊢ (Rel 𝑅 → ((dom 𝑅 ∈ ∪ (𝑅1 “ On) ∧ ran 𝑅 ∈ ∪ (𝑅1 “ On)) → 𝑅 ∈ ∪ (𝑅1 “ On)))
10	3, 9	impbid2 229	1 ⊢ (Rel 𝑅 → (𝑅 ∈ ∪ (𝑅1 “ On) ↔ (dom 𝑅 ∈ ∪ (𝑅1 “ On) ∧ ran 𝑅 ∈ ∪ (𝑅1 “ On))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∈ wcel 2143   ⊆ wss 3906  ∪ cuni 4873   × cxp 5661  dom cdm 5663  ran crn 5664   “ cima 5666  Rel wrel 5668  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: (None)