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Theorem harwdom 9535
Description: The value of the Hartogs function at a set 𝑋 is weakly dominated by 𝒫 (𝑋 Γ— 𝑋). This follows from a more precise analysis of the bound used in hartogs 9488 to prove that (harβ€˜π‘‹) is an ordinal. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
harwdom (𝑋 ∈ 𝑉 β†’ (harβ€˜π‘‹) β‰Ό* 𝒫 (𝑋 Γ— 𝑋))

Proof of Theorem harwdom
Dummy variables 𝑦 π‘Ÿ 𝑓 𝑠 𝑑 𝑀 π‘₯ 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2733 . . . . . 6 {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝑋 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))} = {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝑋 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))}
2 eqid 2733 . . . . . 6 {βŸ¨π‘ , π‘‘βŸ© ∣ βˆƒπ‘€ ∈ 𝑦 βˆƒπ‘§ ∈ 𝑦 ((𝑠 = (π‘“β€˜π‘€) ∧ 𝑑 = (π‘“β€˜π‘§)) ∧ 𝑀 E 𝑧)} = {βŸ¨π‘ , π‘‘βŸ© ∣ βˆƒπ‘€ ∈ 𝑦 βˆƒπ‘§ ∈ 𝑦 ((𝑠 = (π‘“β€˜π‘€) ∧ 𝑑 = (π‘“β€˜π‘§)) ∧ 𝑀 E 𝑧)}
31, 2hartogslem1 9486 . . . . 5 (dom {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝑋 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))} βŠ† 𝒫 (𝑋 Γ— 𝑋) ∧ Fun {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝑋 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))} ∧ (𝑋 ∈ 𝑉 β†’ ran {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝑋 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))} = {π‘₯ ∈ On ∣ π‘₯ β‰Ό 𝑋}))
43simp2i 1141 . . . 4 Fun {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝑋 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))}
53simp1i 1140 . . . . 5 dom {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝑋 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))} βŠ† 𝒫 (𝑋 Γ— 𝑋)
6 sqxpexg 7693 . . . . . 6 (𝑋 ∈ 𝑉 β†’ (𝑋 Γ— 𝑋) ∈ V)
76pwexd 5338 . . . . 5 (𝑋 ∈ 𝑉 β†’ 𝒫 (𝑋 Γ— 𝑋) ∈ V)
8 ssexg 5284 . . . . 5 ((dom {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝑋 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))} βŠ† 𝒫 (𝑋 Γ— 𝑋) ∧ 𝒫 (𝑋 Γ— 𝑋) ∈ V) β†’ dom {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝑋 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))} ∈ V)
95, 7, 8sylancr 588 . . . 4 (𝑋 ∈ 𝑉 β†’ dom {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝑋 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))} ∈ V)
10 funex 7173 . . . 4 ((Fun {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝑋 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))} ∧ dom {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝑋 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))} ∈ V) β†’ {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝑋 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))} ∈ V)
114, 9, 10sylancr 588 . . 3 (𝑋 ∈ 𝑉 β†’ {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝑋 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))} ∈ V)
12 funfn 6535 . . . . . 6 (Fun {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝑋 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))} ↔ {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝑋 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))} Fn dom {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝑋 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))})
134, 12mpbi 229 . . . . 5 {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝑋 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))} Fn dom {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝑋 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))}
1413a1i 11 . . . 4 (𝑋 ∈ 𝑉 β†’ {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝑋 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))} Fn dom {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝑋 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))})
153simp3i 1142 . . . . 5 (𝑋 ∈ 𝑉 β†’ ran {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝑋 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))} = {π‘₯ ∈ On ∣ π‘₯ β‰Ό 𝑋})
16 harval 9504 . . . . 5 (𝑋 ∈ 𝑉 β†’ (harβ€˜π‘‹) = {π‘₯ ∈ On ∣ π‘₯ β‰Ό 𝑋})
1715, 16eqtr4d 2776 . . . 4 (𝑋 ∈ 𝑉 β†’ ran {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝑋 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))} = (harβ€˜π‘‹))
18 df-fo 6506 . . . 4 ({βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝑋 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))}:dom {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝑋 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))}–ontoβ†’(harβ€˜π‘‹) ↔ ({βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝑋 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))} Fn dom {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝑋 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))} ∧ ran {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝑋 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))} = (harβ€˜π‘‹)))
1914, 17, 18sylanbrc 584 . . 3 (𝑋 ∈ 𝑉 β†’ {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝑋 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))}:dom {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝑋 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))}–ontoβ†’(harβ€˜π‘‹))
20 fowdom 9515 . . 3 (({βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝑋 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))} ∈ V ∧ {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝑋 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))}:dom {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝑋 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))}–ontoβ†’(harβ€˜π‘‹)) β†’ (harβ€˜π‘‹) β‰Ό* dom {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝑋 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))})
2111, 19, 20syl2anc 585 . 2 (𝑋 ∈ 𝑉 β†’ (harβ€˜π‘‹) β‰Ό* dom {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝑋 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))})
22 ssdomg 8946 . . . 4 (𝒫 (𝑋 Γ— 𝑋) ∈ V β†’ (dom {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝑋 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))} βŠ† 𝒫 (𝑋 Γ— 𝑋) β†’ dom {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝑋 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))} β‰Ό 𝒫 (𝑋 Γ— 𝑋)))
237, 5, 22mpisyl 21 . . 3 (𝑋 ∈ 𝑉 β†’ dom {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝑋 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))} β‰Ό 𝒫 (𝑋 Γ— 𝑋))
24 domwdom 9518 . . 3 (dom {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝑋 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))} β‰Ό 𝒫 (𝑋 Γ— 𝑋) β†’ dom {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝑋 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))} β‰Ό* 𝒫 (𝑋 Γ— 𝑋))
2523, 24syl 17 . 2 (𝑋 ∈ 𝑉 β†’ dom {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝑋 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))} β‰Ό* 𝒫 (𝑋 Γ— 𝑋))
26 wdomtr 9519 . 2 (((harβ€˜π‘‹) β‰Ό* dom {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝑋 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))} ∧ dom {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝑋 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))} β‰Ό* 𝒫 (𝑋 Γ— 𝑋)) β†’ (harβ€˜π‘‹) β‰Ό* 𝒫 (𝑋 Γ— 𝑋))
2721, 25, 26syl2anc 585 1 (𝑋 ∈ 𝑉 β†’ (harβ€˜π‘‹) β‰Ό* 𝒫 (𝑋 Γ— 𝑋))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3070  {crab 3406  Vcvv 3447   βˆ– cdif 3911   βŠ† wss 3914  π’« cpw 4564   class class class wbr 5109  {copab 5171   I cid 5534   E cep 5540   We wwe 5591   Γ— cxp 5635  dom cdm 5637  ran crn 5638   β†Ύ cres 5639  Oncon0 6321  Fun wfun 6494   Fn wfn 6495  β€“ontoβ†’wfo 6498  β€˜cfv 6500   β‰Ό cdom 8887  OrdIsocoi 9453  harchar 9500   β‰Ό* cwdom 9508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-se 5593  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  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 7317  df-ov 7364  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-en 8890  df-dom 8891  df-sdom 8892  df-oi 9454  df-har 9501  df-wdom 9509
This theorem is referenced by:  gchhar  10623
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