Theorem harwdom 9508

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 9461 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 2737	. . . . . 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 2737	. . . . . 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 9459	. . . . 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 1141	. . . 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 1140	. . . . 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 7710	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → (𝑋 × 𝑋) ∈ V)
7	6	pwexd 5326	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → 𝒫 (𝑋 × 𝑋) ∈ V)
8		ssexg 5270	. . . . 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 588	. . . 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 7175	. . . 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 588	. . 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 6530	. . . . . 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 1142	. . . . 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 9477	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (har‘𝑋) = {𝑥 ∈ On ∣ 𝑥 ≼ 𝑋})
17	15, 16	eqtr4d 2775	. . . 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 6506	. . . 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 584	. . 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 9488	. . 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 585	. 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 8949	. . . 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 9491	. . 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 9492	. 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 585	1 ⊢ (𝑋 ∈ 𝑉 → (har‘𝑋) ≼* 𝒫 (𝑋 × 𝑋))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∃wrex 3062  {crab 3401  Vcvv 3442   ∖ cdif 3900   ⊆ wss 3903  𝒫 cpw 4556   class class class wbr 5100  {copab 5162   I cid 5526   E cep 5531   We wwe 5584   × cxp 5630  dom cdm 5632  ran crn 5633   ↾ cres 5634  Oncon0 6325  Fun wfun 6494   Fn wfn 6495  –onto→wfo 6498  ‘cfv 6500   ≼ cdom 8893  OrdIsocoi 9426  harchar 9473   ≼^* cwdom 9481

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-rep 5226  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-rmo 3352  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-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-se 5586  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-isom 6509  df-riota 7325  df-ov 7371  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-en 8896  df-dom 8897  df-sdom 8898  df-oi 9427  df-har 9474  df-wdom 9482

This theorem is referenced by:  gchhar  10602