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Theorem harwdom 8784
Description: The Hartogs function is weakly dominated by 𝒫 (𝑋 × 𝑋). This follows from a more precise analysis of the bound used in hartogs 8738 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 2778 . . . . . 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 2778 . . . . . 6 {⟨𝑠, 𝑡⟩ ∣ ∃𝑤𝑦𝑧𝑦 ((𝑠 = (𝑓𝑤) ∧ 𝑡 = (𝑓𝑧)) ∧ 𝑤 E 𝑧)} = {⟨𝑠, 𝑡⟩ ∣ ∃𝑤𝑦𝑧𝑦 ((𝑠 = (𝑓𝑤) ∧ 𝑡 = (𝑓𝑧)) ∧ 𝑤 E 𝑧)}
31, 2hartogslem1 8736 . . . . 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 1131 . . . 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 1130 . . . . 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 7241 . . . . . 6 (𝑋𝑉 → (𝑋 × 𝑋) ∈ V)
76pwexd 5091 . . . . 5 (𝑋𝑉 → 𝒫 (𝑋 × 𝑋) ∈ V)
8 ssexg 5041 . . . . 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)
95, 7, 8sylancr 581 . . . 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)
10 funex 6754 . . . 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)
114, 9, 10sylancr 581 . . 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)
12 funfn 6165 . . . . . 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))})
134, 12mpbi 222 . . . . 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))}
1413a1i 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))})
153simp3i 1132 . . . . 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})
16 harval 8756 . . . . 5 (𝑋𝑉 → (har‘𝑋) = {𝑥 ∈ On ∣ 𝑥𝑋})
1715, 16eqtr4d 2817 . . . 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))
18 df-fo 6141 . . . 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)))
1914, 17, 18sylanbrc 578 . . 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))
20 fowdom 8765 . . 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))})
2111, 19, 20syl2anc 579 . 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))})
22 ssdomg 8287 . . . 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)))
237, 5, 22mpisyl 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))
24 domwdom 8768 . . 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))
2523, 24syl 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))
26 wdomtr 8769 . 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))
2721, 25, 26syl2anc 579 1 (𝑋𝑉 → (har‘𝑋) ≼* 𝒫 (𝑋 × 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1071   = wceq 1601  wcel 2107  wrex 3091  {crab 3094  Vcvv 3398  cdif 3789  wss 3792  𝒫 cpw 4379   class class class wbr 4886  {copab 4948   I cid 5260   E cep 5265   We wwe 5313   × cxp 5353  dom cdm 5355  ran crn 5356  cres 5357  Oncon0 5976  Fun wfun 6129   Fn wfn 6130  ontowfo 6133  cfv 6135  cdom 8239  OrdIsocoi 8703  harchar 8750  * cwdom 8751
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-se 5315  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-isom 6144  df-riota 6883  df-wrecs 7689  df-recs 7751  df-er 8026  df-en 8242  df-dom 8243  df-sdom 8244  df-oi 8704  df-har 8752  df-wdom 8753
This theorem is referenced by:  gchhar  9836
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