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Theorem harwdom 8492
Description: The Hartogs function is weakly dominated by 𝒫 (𝑋 × 𝑋). This follows from a more precise analysis of the bound used in hartogs 8446 to prove that (har‘𝑋) is a set. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
harwdom (𝑋𝑉 → (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.
StepHypRef Expression
1 eqid 2621 . . . . . 6 {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} = {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))}
2 eqid 2621 . . . . . 6 {⟨𝑠, 𝑡⟩ ∣ ∃𝑤𝑦𝑧𝑦 ((𝑠 = (𝑓𝑤) ∧ 𝑡 = (𝑓𝑧)) ∧ 𝑤 E 𝑧)} = {⟨𝑠, 𝑡⟩ ∣ ∃𝑤𝑦𝑧𝑦 ((𝑠 = (𝑓𝑤) ∧ 𝑡 = (𝑓𝑧)) ∧ 𝑤 E 𝑧)}
31, 2hartogslem1 8444 . . . . 5 (dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} ⊆ 𝒫 (𝑋 × 𝑋) ∧ Fun {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} ∧ (𝑋𝑉 → ran {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} = {𝑥 ∈ On ∣ 𝑥𝑋}))
43simp2i 1070 . . . 4 Fun {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))}
53simp1i 1069 . . . . 5 dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} ⊆ 𝒫 (𝑋 × 𝑋)
6 sqxpexg 6960 . . . . . 6 (𝑋𝑉 → (𝑋 × 𝑋) ∈ V)
7 pwexg 4848 . . . . . 6 ((𝑋 × 𝑋) ∈ V → 𝒫 (𝑋 × 𝑋) ∈ V)
86, 7syl 17 . . . . 5 (𝑋𝑉 → 𝒫 (𝑋 × 𝑋) ∈ V)
9 ssexg 4802 . . . . 5 ((dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (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 𝑟𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} ∈ V)
105, 8, 9sylancr 695 . . . 4 (𝑋𝑉 → dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} ∈ V)
11 funex 6479 . . . 4 ((Fun {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} ∧ dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} ∈ V) → {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} ∈ V)
124, 10, 11sylancr 695 . . 3 (𝑋𝑉 → {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} ∈ V)
13 funfn 5916 . . . . . 6 (Fun {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} ↔ {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} Fn dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))})
144, 13mpbi 220 . . . . 5 {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} Fn dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))}
1514a1i 11 . . . 4 (𝑋𝑉 → {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} Fn dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))})
163simp3i 1071 . . . . 5 (𝑋𝑉 → ran {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} = {𝑥 ∈ On ∣ 𝑥𝑋})
17 harval 8464 . . . . 5 (𝑋𝑉 → (har‘𝑋) = {𝑥 ∈ On ∣ 𝑥𝑋})
1816, 17eqtr4d 2658 . . . 4 (𝑋𝑉 → ran {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} = (har‘𝑋))
19 df-fo 5892 . . . 4 ({⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))}:dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))}–onto→(har‘𝑋) ↔ ({⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} Fn dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} ∧ ran {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} = (har‘𝑋)))
2015, 18, 19sylanbrc 698 . . 3 (𝑋𝑉 → {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))}:dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))}–onto→(har‘𝑋))
21 fowdom 8473 . . 3 (({⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} ∈ V ∧ {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))}:dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (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 𝑟𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))})
2212, 20, 21syl2anc 693 . 2 (𝑋𝑉 → (har‘𝑋) ≼* dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))})
23 ssdomg 7998 . . . 4 (𝒫 (𝑋 × 𝑋) ∈ V → (dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} ⊆ 𝒫 (𝑋 × 𝑋) → dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} ≼ 𝒫 (𝑋 × 𝑋)))
248, 5, 23mpisyl 21 . . 3 (𝑋𝑉 → dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} ≼ 𝒫 (𝑋 × 𝑋))
25 domwdom 8476 . . 3 (dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} ≼ 𝒫 (𝑋 × 𝑋) → dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} ≼* 𝒫 (𝑋 × 𝑋))
2624, 25syl 17 . 2 (𝑋𝑉 → dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} ≼* 𝒫 (𝑋 × 𝑋))
27 wdomtr 8477 . 2 (((har‘𝑋) ≼* dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} ∧ dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝑋 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} ≼* 𝒫 (𝑋 × 𝑋)) → (har‘𝑋) ≼* 𝒫 (𝑋 × 𝑋))
2822, 26, 27syl2anc 693 1 (𝑋𝑉 → (har‘𝑋) ≼* 𝒫 (𝑋 × 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1037   = wceq 1482  wcel 1989  wrex 2912  {crab 2915  Vcvv 3198  cdif 3569  wss 3572  𝒫 cpw 4156   class class class wbr 4651  {copab 4710   I cid 5021   E cep 5026   We wwe 5070   × cxp 5110  dom cdm 5112  ran crn 5113  cres 5114  Oncon0 5721  Fun wfun 5880   Fn wfn 5881  ontowfo 5884  cfv 5886  cdom 7950  OrdIsocoi 8411  harchar 8458  * cwdom 8459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rmo 2919  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-se 5072  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-isom 5895  df-riota 6608  df-wrecs 7404  df-recs 7465  df-er 7739  df-en 7953  df-dom 7954  df-sdom 7955  df-oi 8412  df-har 8460  df-wdom 8461
This theorem is referenced by:  gchhar  9498
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