Theorem relwf 45418

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 45416	. . 3 ⊢ (𝑅 ∈ ∪ (𝑅1 “ On) → dom 𝑅 ∈ ∪ (𝑅1 “ On))
2		rnwf 45417	. . 3 ⊢ (𝑅 ∈ ∪ (𝑅1 “ On) → ran 𝑅 ∈ ∪ (𝑅1 “ On))
3	1, 2	jca 516	. 2 ⊢ (𝑅 ∈ ∪ (𝑅1 “ On) → (dom 𝑅 ∈ ∪ (𝑅1 “ On) ∧ ran 𝑅 ∈ ∪ (𝑅1 “ On)))
4		xpwf 45415	. . 3 ⊢ ((dom 𝑅 ∈ ∪ (𝑅1 “ On) ∧ ran 𝑅 ∈ ∪ (𝑅1 “ On)) → (dom 𝑅 × ran 𝑅) ∈ ∪ (𝑅1 “ On))
5		relssdmrn 6227	. . . . 5 ⊢ (Rel 𝑅 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅))
6		sswf 9730	. . . . 5 ⊢ (((dom 𝑅 × ran 𝑅) ∈ ∪ (𝑅1 “ On) ∧ 𝑅 ⊆ (dom 𝑅 × ran 𝑅)) → 𝑅 ∈ ∪ (𝑅1 “ On))
7	5, 6	sylan2 599	. . . 4 ⊢ (((dom 𝑅 × ran 𝑅) ∈ ∪ (𝑅1 “ On) ∧ Rel 𝑅) → 𝑅 ∈ ∪ (𝑅1 “ On))
8	7	expcom 414	. . 3 ⊢ (Rel 𝑅 → ((dom 𝑅 × ran 𝑅) ∈ ∪ (𝑅1 “ On) → 𝑅 ∈ ∪ (𝑅1 “ On)))
9	4, 8	syl5 34	. 2 ⊢ (Rel 𝑅 → ((dom 𝑅 ∈ ∪ (𝑅1 “ On) ∧ ran 𝑅 ∈ ∪ (𝑅1 “ On)) → 𝑅 ∈ ∪ (𝑅1 “ On)))
10	3, 9	impbid2 227	1 ⊢ (Rel 𝑅 → (𝑅 ∈ ∪ (𝑅1 “ On) ↔ (dom 𝑅 ∈ ∪ (𝑅1 “ On) ∧ ran 𝑅 ∈ ∪ (𝑅1 “ On))))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∈ wcel 2119   ⊆ wss 3890  ∪ cuni 4845   × cxp 5623  dom cdm 5625  ran crn 5626   “ cima 5628  Rel wrel 5630  Oncon0 6317  𝑅₁cr1 9684

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7366  df-om 7814  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-r1 9686  df-rank 9687

This theorem is referenced by: (None)