Theorem relwf 46221

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 46219	. . 3 ⊢ (𝑅 ∈ ∪ (𝑅1 “ On) → dom 𝑅 ∈ ∪ (𝑅1 “ On))
2		rnwf 46220	. . 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 46218	. . 3 ⊢ ((dom 𝑅 ∈ ∪ (𝑅1 “ On) ∧ ran 𝑅 ∈ ∪ (𝑅1 “ On)) → (dom 𝑅 × ran 𝑅) ∈ ∪ (𝑅1 “ On))
5		relssdmrn 6295	. . . . 5 ⊢ (Rel 𝑅 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅))
6		sswf 9853	. . . . 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 2147   ⊆ wss 3900  ∪ cuni 4872   × cxp 5656  dom cdm 5658  ran crn 5659   “ cima 5661  Rel wrel 5663  Oncon0 6388  𝑅₁cr1 9803

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-om 7912  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-r1 9805  df-rank 9806

This theorem is used by: (None)