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Theorem harwdom 9608
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 9561 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 2728 . . . . . 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 2728 . . . . . 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 9559 . . . . 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 1138 . . . 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 1137 . . . . 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 7751 . . . . . 6 (𝑋 ∈ 𝑉 β†’ (𝑋 Γ— 𝑋) ∈ V)
76pwexd 5373 . . . . 5 (𝑋 ∈ 𝑉 β†’ 𝒫 (𝑋 Γ— 𝑋) ∈ V)
8 ssexg 5317 . . . . 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 7225 . . . 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 6577 . . . . . 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 1139 . . . . 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 9577 . . . . 5 (𝑋 ∈ 𝑉 β†’ (harβ€˜π‘‹) = {π‘₯ ∈ On ∣ π‘₯ β‰Ό 𝑋})
1715, 16eqtr4d 2771 . . . 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 6548 . . . 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 9588 . . 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 9014 . . . 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 9591 . . 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 9592 . 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 1085   = wceq 1534   ∈ wcel 2099  βˆƒwrex 3066  {crab 3428  Vcvv 3470   βˆ– cdif 3942   βŠ† wss 3945  π’« cpw 4598   class class class wbr 5142  {copab 5204   I cid 5569   E cep 5575   We wwe 5626   Γ— cxp 5670  dom cdm 5672  ran crn 5673   β†Ύ cres 5674  Oncon0 6363  Fun wfun 6536   Fn wfn 6537  β€“ontoβ†’wfo 6540  β€˜cfv 6542   β‰Ό cdom 8955  OrdIsocoi 9526  harchar 9573   β‰Ό* cwdom 9581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-en 8958  df-dom 8959  df-sdom 8960  df-oi 9527  df-har 9574  df-wdom 9582
This theorem is referenced by:  gchhar  10696
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