Theorem relwf 45717

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 45715	. . 3 ⊢ (𝑅 ∈ ∪ (𝑅1 “ On) → dom 𝑅 ∈ ∪ (𝑅1 “ On))
2		rnwf 45716	. . 3 ⊢ (𝑅 ∈ ∪ (𝑅1 “ On) → ran 𝑅 ∈ ∪ (𝑅1 “ On))
3	1, 2	jca 521	. 2 ⊢ (𝑅 ∈ ∪ (𝑅1 “ On) → (dom 𝑅 ∈ ∪ (𝑅1 “ On) ∧ ran 𝑅 ∈ ∪ (𝑅1 “ On)))
4		xpwf 45714	. . 3 ⊢ ((dom 𝑅 ∈ ∪ (𝑅1 “ On) ∧ ran 𝑅 ∈ ∪ (𝑅1 “ On)) → (dom 𝑅 × ran 𝑅) ∈ ∪ (𝑅1 “ On))
5		relssdmrn 6276	. . . . 5 ⊢ (Rel 𝑅 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅))
6		sswf 9790	. . . . 5 ⊢ (((dom 𝑅 × ran 𝑅) ∈ ∪ (𝑅1 “ On) ∧ 𝑅 ⊆ (dom 𝑅 × ran 𝑅)) → 𝑅 ∈ ∪ (𝑅1 “ On))
7	5, 6	sylan2 605	. . . 4 ⊢ (((dom 𝑅 × ran 𝑅) ∈ ∪ (𝑅1 “ On) ∧ Rel 𝑅) → 𝑅 ∈ ∪ (𝑅1 “ On))
8	7	expcom 419	. . 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))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2146   ⊆ wss 3908  ∪ cuni 4877   × cxp 5664  dom cdm 5666  ran crn 5667   “ cima 5669  Rel wrel 5671  Oncon0 6367  𝑅₁cr1 9744

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

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 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-int 4918  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7426  df-om 7872  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-r1 9746  df-rank 9747

This theorem is used by: (None)