Theorem harwdom 9483

Description: The value of the Hartogs function at a set 𝑋 is weakly dominated by 𝒫 (𝑋 × 𝑋). This follows from a more precise analysis of the bound used in hartogs 9436 to prove that (har‘𝑋) is an ordinal. (Contributed by Mario Carneiro, 15-May-2015.)

Assertion

Ref	Expression
harwdom	⊢ (𝑋 ∈ 𝑉 → (har‘𝑋) ≼* 𝒫 (𝑋 × 𝑋))

Proof of Theorem harwdom

Dummy variables 𝑦 𝑟 𝑓 𝑠 𝑡 𝑤 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2729	. . . . . 6 ⊢ {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} = {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))}
2		eqid 2729	. . . . . 6 ⊢ {⟨𝑠, 𝑡⟩ ∣ ∃𝑤 ∈ 𝑦 ∃𝑧 ∈ 𝑦 ((𝑠 = (𝑓‘𝑤) ∧ 𝑡 = (𝑓‘𝑧)) ∧ 𝑤 E 𝑧)} = {⟨𝑠, 𝑡⟩ ∣ ∃𝑤 ∈ 𝑦 ∃𝑧 ∈ 𝑦 ((𝑠 = (𝑓‘𝑤) ∧ 𝑡 = (𝑓‘𝑧)) ∧ 𝑤 E 𝑧)}
3	1, 2	hartogslem1 9434	. . . . 5 ⊢ (dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} ⊆ 𝒫 (𝑋 × 𝑋) ∧ Fun {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} ∧ (𝑋 ∈ 𝑉 → ran {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} = {𝑥 ∈ On ∣ 𝑥 ≼ 𝑋}))
4	3	simp2i 1140	. . . 4 ⊢ Fun {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))}
5	3	simp1i 1139	. . . . 5 ⊢ dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} ⊆ 𝒫 (𝑋 × 𝑋)
6		sqxpexg 7691	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → (𝑋 × 𝑋) ∈ V)
7	6	pwexd 5318	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → 𝒫 (𝑋 × 𝑋) ∈ V)
8		ssexg 5262	. . . . 5 ⊢ ((dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} ⊆ 𝒫 (𝑋 × 𝑋) ∧ 𝒫 (𝑋 × 𝑋) ∈ V) → dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} ∈ V)
9	5, 7, 8	sylancr 587	. . . 4 ⊢ (𝑋 ∈ 𝑉 → dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} ∈ V)
10		funex 7155	. . . 4 ⊢ ((Fun {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} ∧ dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} ∈ V) → {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} ∈ V)
11	4, 9, 10	sylancr 587	. . 3 ⊢ (𝑋 ∈ 𝑉 → {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} ∈ V)
12		funfn 6512	. . . . . 6 ⊢ (Fun {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} ↔ {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} Fn dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))})
13	4, 12	mpbi 230	. . . . 5 ⊢ {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} Fn dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))}
14	13	a1i 11	. . . 4 ⊢ (𝑋 ∈ 𝑉 → {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} Fn dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))})
15	3	simp3i 1141	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → ran {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} = {𝑥 ∈ On ∣ 𝑥 ≼ 𝑋})
16		harval 9452	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (har‘𝑋) = {𝑥 ∈ On ∣ 𝑥 ≼ 𝑋})
17	15, 16	eqtr4d 2767	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ran {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} = (har‘𝑋))
18		df-fo 6488	. . . 4 ⊢ ({⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))}:dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))}–onto→(har‘𝑋) ↔ ({⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} Fn dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} ∧ ran {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} = (har‘𝑋)))
19	14, 17, 18	sylanbrc 583	. . 3 ⊢ (𝑋 ∈ 𝑉 → {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))}:dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))}–onto→(har‘𝑋))
20		fowdom 9463	. . 3 ⊢ (({⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} ∈ V ∧ {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))}:dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))}–onto→(har‘𝑋)) → (har‘𝑋) ≼* dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))})
21	11, 19, 20	syl2anc 584	. 2 ⊢ (𝑋 ∈ 𝑉 → (har‘𝑋) ≼* dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))})
22		ssdomg 8925	. . . 4 ⊢ (𝒫 (𝑋 × 𝑋) ∈ V → (dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} ⊆ 𝒫 (𝑋 × 𝑋) → dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} ≼ 𝒫 (𝑋 × 𝑋)))
23	7, 5, 22	mpisyl 21	. . 3 ⊢ (𝑋 ∈ 𝑉 → dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} ≼ 𝒫 (𝑋 × 𝑋))
24		domwdom 9466	. . 3 ⊢ (dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} ≼ 𝒫 (𝑋 × 𝑋) → dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} ≼* 𝒫 (𝑋 × 𝑋))
25	23, 24	syl 17	. 2 ⊢ (𝑋 ∈ 𝑉 → dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} ≼* 𝒫 (𝑋 × 𝑋))
26		wdomtr 9467	. 2 ⊢ (((har‘𝑋) ≼* dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} ∧ dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} ≼* 𝒫 (𝑋 × 𝑋)) → (har‘𝑋) ≼* 𝒫 (𝑋 × 𝑋))
27	21, 25, 26	syl2anc 584	1 ⊢ (𝑋 ∈ 𝑉 → (har‘𝑋) ≼* 𝒫 (𝑋 × 𝑋))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  {crab 3394  Vcvv 3436   ∖ cdif 3900   ⊆ wss 3903  𝒫 cpw 4551   class class class wbr 5092  {copab 5154   I cid 5513   E cep 5518   We wwe 5571   × cxp 5617  dom cdm 5619  ran crn 5620   ↾ cres 5621  Oncon0 6307  Fun wfun 6476   Fn wfn 6477  –onto→wfo 6480  ‘cfv 6482   ≼ cdom 8870  OrdIsocoi 9401  harchar 9448   ≼^* cwdom 9456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-en 8873  df-dom 8874  df-sdom 8875  df-oi 9402  df-har 9449  df-wdom 9457

This theorem is referenced by:  gchhar  10573