Theorem relwf 45317

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 45315	. . 3 ⊢ (𝑅 ∈ ∪ (𝑅1 “ On) → dom 𝑅 ∈ ∪ (𝑅1 “ On))
2		rnwf 45316	. . 3 ⊢ (𝑅 ∈ ∪ (𝑅1 “ On) → ran 𝑅 ∈ ∪ (𝑅1 “ On))
3	1, 2	jca 511	. 2 ⊢ (𝑅 ∈ ∪ (𝑅1 “ On) → (dom 𝑅 ∈ ∪ (𝑅1 “ On) ∧ ran 𝑅 ∈ ∪ (𝑅1 “ On)))
4		xpwf 45314	. . 3 ⊢ ((dom 𝑅 ∈ ∪ (𝑅1 “ On) ∧ ran 𝑅 ∈ ∪ (𝑅1 “ On)) → (dom 𝑅 × ran 𝑅) ∈ ∪ (𝑅1 “ On))
5		relssdmrn 6235	. . . . 5 ⊢ (Rel 𝑅 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅))
6		sswf 9732	. . . . 5 ⊢ (((dom 𝑅 × ran 𝑅) ∈ ∪ (𝑅1 “ On) ∧ 𝑅 ⊆ (dom 𝑅 × ran 𝑅)) → 𝑅 ∈ ∪ (𝑅1 “ On))
7	5, 6	sylan2 594	. . . 4 ⊢ (((dom 𝑅 × ran 𝑅) ∈ ∪ (𝑅1 “ On) ∧ Rel 𝑅) → 𝑅 ∈ ∪ (𝑅1 “ On))
8	7	expcom 413	. . 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 226	1 ⊢ (Rel 𝑅 → (𝑅 ∈ ∪ (𝑅1 “ On) ↔ (dom 𝑅 ∈ ∪ (𝑅1 “ On) ∧ ran 𝑅 ∈ ∪ (𝑅1 “ On))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2114   ⊆ wss 3903  ∪ cuni 4865   × cxp 5630  dom cdm 5632  ran crn 5633   “ cima 5635  Rel wrel 5637  Oncon0 6325  𝑅₁cr1 9686

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-om 7819  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-r1 9688  df-rank 9689

This theorem is referenced by: (None)