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Theorem harwdom 9585
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 9538 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 2732 . . . . . 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 2732 . . . . . 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 9536 . . . . 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 1140 . . . 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 1139 . . . . 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 7741 . . . . . 6 (𝑋 ∈ 𝑉 β†’ (𝑋 Γ— 𝑋) ∈ V)
76pwexd 5377 . . . . 5 (𝑋 ∈ 𝑉 β†’ 𝒫 (𝑋 Γ— 𝑋) ∈ V)
8 ssexg 5323 . . . . 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 587 . . . 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 7220 . . . 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 587 . . 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 6578 . . . . . 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 1141 . . . . 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 9554 . . . . 5 (𝑋 ∈ 𝑉 β†’ (harβ€˜π‘‹) = {π‘₯ ∈ On ∣ π‘₯ β‰Ό 𝑋})
1715, 16eqtr4d 2775 . . . 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 6549 . . . 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 583 . . 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 9565 . . 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 584 . 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 8995 . . . 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 9568 . . 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 9569 . 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 584 1 (𝑋 ∈ 𝑉 β†’ (harβ€˜π‘‹) β‰Ό* 𝒫 (𝑋 Γ— 𝑋))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3070  {crab 3432  Vcvv 3474   βˆ– cdif 3945   βŠ† wss 3948  π’« cpw 4602   class class class wbr 5148  {copab 5210   I cid 5573   E cep 5579   We wwe 5630   Γ— cxp 5674  dom cdm 5676  ran crn 5677   β†Ύ cres 5678  Oncon0 6364  Fun wfun 6537   Fn wfn 6538  β€“ontoβ†’wfo 6541  β€˜cfv 6543   β‰Ό cdom 8936  OrdIsocoi 9503  harchar 9550   β‰Ό* cwdom 9558
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7364  df-ov 7411  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-en 8939  df-dom 8940  df-sdom 8941  df-oi 9504  df-har 9551  df-wdom 9559
This theorem is referenced by:  gchhar  10673
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