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Theorem harwdom 9583
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 9536 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 2724 . . . . . 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 2724 . . . . . 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 9534 . . . . 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 1137 . . . 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 1136 . . . . 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 7736 . . . . . 6 (𝑋 ∈ 𝑉 β†’ (𝑋 Γ— 𝑋) ∈ V)
76pwexd 5368 . . . . 5 (𝑋 ∈ 𝑉 β†’ 𝒫 (𝑋 Γ— 𝑋) ∈ V)
8 ssexg 5314 . . . . 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 586 . . . 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 7213 . . . 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 586 . . 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 6569 . . . . . 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 1138 . . . . 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 9552 . . . . 5 (𝑋 ∈ 𝑉 β†’ (harβ€˜π‘‹) = {π‘₯ ∈ On ∣ π‘₯ β‰Ό 𝑋})
1715, 16eqtr4d 2767 . . . 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 6540 . . . 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 582 . . 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 9563 . . 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 583 . 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 8993 . . . 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 9566 . . 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 9567 . 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 583 1 (𝑋 ∈ 𝑉 β†’ (harβ€˜π‘‹) β‰Ό* 𝒫 (𝑋 Γ— 𝑋))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3062  {crab 3424  Vcvv 3466   βˆ– cdif 3938   βŠ† wss 3941  π’« cpw 4595   class class class wbr 5139  {copab 5201   I cid 5564   E cep 5570   We wwe 5621   Γ— cxp 5665  dom cdm 5667  ran crn 5668   β†Ύ cres 5669  Oncon0 6355  Fun wfun 6528   Fn wfn 6529  β€“ontoβ†’wfo 6532  β€˜cfv 6534   β‰Ό cdom 8934  OrdIsocoi 9501  harchar 9548   β‰Ό* cwdom 9556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-se 5623  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-isom 6543  df-riota 7358  df-ov 7405  df-2nd 7970  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-en 8937  df-dom 8938  df-sdom 8939  df-oi 9502  df-har 9549  df-wdom 9557
This theorem is referenced by:  gchhar  10671
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