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Theorem dfrecs2 33308
Description: A quantifier-free definition of recs. (Contributed by Scott Fenton, 17-Jul-2020.)
Assertion
Ref Expression
dfrecs2 recs(𝐹) = (( Funs ∩ (Domain “ On)) ∖ dom (( E ∘ Domain) ∖ Fix (Apply ∘ (FullFun𝐹 ∘ Restrict))))

Proof of Theorem dfrecs2
Dummy variables 𝑓 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfrecs3 7998 . 2 recs(𝐹) = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
2 elin 4166 . . . . . . . . 9 (𝑓 ∈ ( Funs ∩ (Domain “ On)) ↔ (𝑓 Funs 𝑓 ∈ (Domain “ On)))
3 vex 3495 . . . . . . . . . . 11 𝑓 ∈ V
43elfuns 33273 . . . . . . . . . 10 (𝑓 Funs ↔ Fun 𝑓)
5 vex 3495 . . . . . . . . . . . . . 14 𝑥 ∈ V
65, 3brcnv 5746 . . . . . . . . . . . . 13 (𝑥Domain𝑓𝑓Domain𝑥)
73, 5brdomain 33291 . . . . . . . . . . . . 13 (𝑓Domain𝑥𝑥 = dom 𝑓)
86, 7bitri 276 . . . . . . . . . . . 12 (𝑥Domain𝑓𝑥 = dom 𝑓)
98rexbii 3244 . . . . . . . . . . 11 (∃𝑥 ∈ On 𝑥Domain𝑓 ↔ ∃𝑥 ∈ On 𝑥 = dom 𝑓)
103elima 5927 . . . . . . . . . . 11 (𝑓 ∈ (Domain “ On) ↔ ∃𝑥 ∈ On 𝑥Domain𝑓)
11 risset 3264 . . . . . . . . . . 11 (dom 𝑓 ∈ On ↔ ∃𝑥 ∈ On 𝑥 = dom 𝑓)
129, 10, 113bitr4i 304 . . . . . . . . . 10 (𝑓 ∈ (Domain “ On) ↔ dom 𝑓 ∈ On)
134, 12anbi12i 626 . . . . . . . . 9 ((𝑓 Funs 𝑓 ∈ (Domain “ On)) ↔ (Fun 𝑓 ∧ dom 𝑓 ∈ On))
142, 13bitri 276 . . . . . . . 8 (𝑓 ∈ ( Funs ∩ (Domain “ On)) ↔ (Fun 𝑓 ∧ dom 𝑓 ∈ On))
153eldm 5762 . . . . . . . . . . 11 (𝑓 ∈ dom (( E ∘ Domain) ∖ Fix (Apply ∘ (FullFun𝐹 ∘ Restrict))) ↔ ∃𝑦 𝑓(( E ∘ Domain) ∖ Fix (Apply ∘ (FullFun𝐹 ∘ Restrict)))𝑦)
16 brdif 5110 . . . . . . . . . . . . 13 (𝑓(( E ∘ Domain) ∖ Fix (Apply ∘ (FullFun𝐹 ∘ Restrict)))𝑦 ↔ (𝑓( E ∘ Domain)𝑦 ∧ ¬ 𝑓 Fix (Apply ∘ (FullFun𝐹 ∘ Restrict))𝑦))
17 vex 3495 . . . . . . . . . . . . . . . 16 𝑦 ∈ V
183, 17brco 5734 . . . . . . . . . . . . . . 15 (𝑓( E ∘ Domain)𝑦 ↔ ∃𝑥(𝑓Domain𝑥𝑥 E 𝑦))
197anbi1i 623 . . . . . . . . . . . . . . . . 17 ((𝑓Domain𝑥𝑥 E 𝑦) ↔ (𝑥 = dom 𝑓𝑥 E 𝑦))
2019exbii 1839 . . . . . . . . . . . . . . . 16 (∃𝑥(𝑓Domain𝑥𝑥 E 𝑦) ↔ ∃𝑥(𝑥 = dom 𝑓𝑥 E 𝑦))
213dmex 7605 . . . . . . . . . . . . . . . . 17 dom 𝑓 ∈ V
22 breq1 5060 . . . . . . . . . . . . . . . . 17 (𝑥 = dom 𝑓 → (𝑥 E 𝑦 ↔ dom 𝑓 E 𝑦))
2321, 22ceqsexv 3539 . . . . . . . . . . . . . . . 16 (∃𝑥(𝑥 = dom 𝑓𝑥 E 𝑦) ↔ dom 𝑓 E 𝑦)
2420, 23bitri 276 . . . . . . . . . . . . . . 15 (∃𝑥(𝑓Domain𝑥𝑥 E 𝑦) ↔ dom 𝑓 E 𝑦)
2521, 17brcnv 5746 . . . . . . . . . . . . . . . 16 (dom 𝑓 E 𝑦𝑦 E dom 𝑓)
2621epeli 5461 . . . . . . . . . . . . . . . 16 (𝑦 E dom 𝑓𝑦 ∈ dom 𝑓)
2725, 26bitri 276 . . . . . . . . . . . . . . 15 (dom 𝑓 E 𝑦𝑦 ∈ dom 𝑓)
2818, 24, 273bitri 298 . . . . . . . . . . . . . 14 (𝑓( E ∘ Domain)𝑦𝑦 ∈ dom 𝑓)
29 df-br 5058 . . . . . . . . . . . . . . . 16 (𝑓 Fix (Apply ∘ (FullFun𝐹 ∘ Restrict))𝑦 ↔ ⟨𝑓, 𝑦⟩ ∈ Fix (Apply ∘ (FullFun𝐹 ∘ Restrict)))
30 opex 5347 . . . . . . . . . . . . . . . . 17 𝑓, 𝑦⟩ ∈ V
3130elfix 33261 . . . . . . . . . . . . . . . 16 (⟨𝑓, 𝑦⟩ ∈ Fix (Apply ∘ (FullFun𝐹 ∘ Restrict)) ↔ ⟨𝑓, 𝑦⟩(Apply ∘ (FullFun𝐹 ∘ Restrict))⟨𝑓, 𝑦⟩)
3230, 30brco 5734 . . . . . . . . . . . . . . . . 17 (⟨𝑓, 𝑦⟩(Apply ∘ (FullFun𝐹 ∘ Restrict))⟨𝑓, 𝑦⟩ ↔ ∃𝑥(⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥𝑥Apply⟨𝑓, 𝑦⟩))
33 ancom 461 . . . . . . . . . . . . . . . . . . . 20 ((⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥𝑥Apply⟨𝑓, 𝑦⟩) ↔ (𝑥Apply⟨𝑓, 𝑦⟩ ∧ ⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥))
345, 30brcnv 5746 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥Apply⟨𝑓, 𝑦⟩ ↔ ⟨𝑓, 𝑦⟩Apply𝑥)
353, 17, 5brapply 33296 . . . . . . . . . . . . . . . . . . . . . 22 (⟨𝑓, 𝑦⟩Apply𝑥𝑥 = (𝑓𝑦))
3634, 35bitri 276 . . . . . . . . . . . . . . . . . . . . 21 (𝑥Apply⟨𝑓, 𝑦⟩ ↔ 𝑥 = (𝑓𝑦))
3736anbi1i 623 . . . . . . . . . . . . . . . . . . . 20 ((𝑥Apply⟨𝑓, 𝑦⟩ ∧ ⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥) ↔ (𝑥 = (𝑓𝑦) ∧ ⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥))
3833, 37bitri 276 . . . . . . . . . . . . . . . . . . 19 ((⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥𝑥Apply⟨𝑓, 𝑦⟩) ↔ (𝑥 = (𝑓𝑦) ∧ ⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥))
3938exbii 1839 . . . . . . . . . . . . . . . . . 18 (∃𝑥(⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥𝑥Apply⟨𝑓, 𝑦⟩) ↔ ∃𝑥(𝑥 = (𝑓𝑦) ∧ ⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥))
40 fvex 6676 . . . . . . . . . . . . . . . . . . 19 (𝑓𝑦) ∈ V
41 breq2 5061 . . . . . . . . . . . . . . . . . . 19 (𝑥 = (𝑓𝑦) → (⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥 ↔ ⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)(𝑓𝑦)))
4240, 41ceqsexv 3539 . . . . . . . . . . . . . . . . . 18 (∃𝑥(𝑥 = (𝑓𝑦) ∧ ⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥) ↔ ⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)(𝑓𝑦))
4339, 42bitri 276 . . . . . . . . . . . . . . . . 17 (∃𝑥(⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥𝑥Apply⟨𝑓, 𝑦⟩) ↔ ⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)(𝑓𝑦))
4430, 40brco 5734 . . . . . . . . . . . . . . . . . 18 (⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)(𝑓𝑦) ↔ ∃𝑥(⟨𝑓, 𝑦⟩Restrict𝑥𝑥FullFun𝐹(𝑓𝑦)))
453, 17, 5brrestrict 33307 . . . . . . . . . . . . . . . . . . . . 21 (⟨𝑓, 𝑦⟩Restrict𝑥𝑥 = (𝑓𝑦))
4645anbi1i 623 . . . . . . . . . . . . . . . . . . . 20 ((⟨𝑓, 𝑦⟩Restrict𝑥𝑥FullFun𝐹(𝑓𝑦)) ↔ (𝑥 = (𝑓𝑦) ∧ 𝑥FullFun𝐹(𝑓𝑦)))
4746exbii 1839 . . . . . . . . . . . . . . . . . . 19 (∃𝑥(⟨𝑓, 𝑦⟩Restrict𝑥𝑥FullFun𝐹(𝑓𝑦)) ↔ ∃𝑥(𝑥 = (𝑓𝑦) ∧ 𝑥FullFun𝐹(𝑓𝑦)))
483resex 5892 . . . . . . . . . . . . . . . . . . . 20 (𝑓𝑦) ∈ V
49 breq1 5060 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = (𝑓𝑦) → (𝑥FullFun𝐹(𝑓𝑦) ↔ (𝑓𝑦)FullFun𝐹(𝑓𝑦)))
5048, 49ceqsexv 3539 . . . . . . . . . . . . . . . . . . 19 (∃𝑥(𝑥 = (𝑓𝑦) ∧ 𝑥FullFun𝐹(𝑓𝑦)) ↔ (𝑓𝑦)FullFun𝐹(𝑓𝑦))
5147, 50bitri 276 . . . . . . . . . . . . . . . . . 18 (∃𝑥(⟨𝑓, 𝑦⟩Restrict𝑥𝑥FullFun𝐹(𝑓𝑦)) ↔ (𝑓𝑦)FullFun𝐹(𝑓𝑦))
5248, 40brfullfun 33306 . . . . . . . . . . . . . . . . . 18 ((𝑓𝑦)FullFun𝐹(𝑓𝑦) ↔ (𝑓𝑦) = (𝐹‘(𝑓𝑦)))
5344, 51, 523bitri 298 . . . . . . . . . . . . . . . . 17 (⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)(𝑓𝑦) ↔ (𝑓𝑦) = (𝐹‘(𝑓𝑦)))
5432, 43, 533bitri 298 . . . . . . . . . . . . . . . 16 (⟨𝑓, 𝑦⟩(Apply ∘ (FullFun𝐹 ∘ Restrict))⟨𝑓, 𝑦⟩ ↔ (𝑓𝑦) = (𝐹‘(𝑓𝑦)))
5529, 31, 543bitri 298 . . . . . . . . . . . . . . 15 (𝑓 Fix (Apply ∘ (FullFun𝐹 ∘ Restrict))𝑦 ↔ (𝑓𝑦) = (𝐹‘(𝑓𝑦)))
5655notbii 321 . . . . . . . . . . . . . 14 𝑓 Fix (Apply ∘ (FullFun𝐹 ∘ Restrict))𝑦 ↔ ¬ (𝑓𝑦) = (𝐹‘(𝑓𝑦)))
5728, 56anbi12i 626 . . . . . . . . . . . . 13 ((𝑓( E ∘ Domain)𝑦 ∧ ¬ 𝑓 Fix (Apply ∘ (FullFun𝐹 ∘ Restrict))𝑦) ↔ (𝑦 ∈ dom 𝑓 ∧ ¬ (𝑓𝑦) = (𝐹‘(𝑓𝑦))))
5816, 57bitri 276 . . . . . . . . . . . 12 (𝑓(( E ∘ Domain) ∖ Fix (Apply ∘ (FullFun𝐹 ∘ Restrict)))𝑦 ↔ (𝑦 ∈ dom 𝑓 ∧ ¬ (𝑓𝑦) = (𝐹‘(𝑓𝑦))))
5958exbii 1839 . . . . . . . . . . 11 (∃𝑦 𝑓(( E ∘ Domain) ∖ Fix (Apply ∘ (FullFun𝐹 ∘ Restrict)))𝑦 ↔ ∃𝑦(𝑦 ∈ dom 𝑓 ∧ ¬ (𝑓𝑦) = (𝐹‘(𝑓𝑦))))
6015, 59bitri 276 . . . . . . . . . 10 (𝑓 ∈ dom (( E ∘ Domain) ∖ Fix (Apply ∘ (FullFun𝐹 ∘ Restrict))) ↔ ∃𝑦(𝑦 ∈ dom 𝑓 ∧ ¬ (𝑓𝑦) = (𝐹‘(𝑓𝑦))))
61 df-rex 3141 . . . . . . . . . 10 (∃𝑦 ∈ dom 𝑓 ¬ (𝑓𝑦) = (𝐹‘(𝑓𝑦)) ↔ ∃𝑦(𝑦 ∈ dom 𝑓 ∧ ¬ (𝑓𝑦) = (𝐹‘(𝑓𝑦))))
62 rexnal 3235 . . . . . . . . . 10 (∃𝑦 ∈ dom 𝑓 ¬ (𝑓𝑦) = (𝐹‘(𝑓𝑦)) ↔ ¬ ∀𝑦 ∈ dom 𝑓(𝑓𝑦) = (𝐹‘(𝑓𝑦)))
6360, 61, 623bitr2ri 301 . . . . . . . . 9 (¬ ∀𝑦 ∈ dom 𝑓(𝑓𝑦) = (𝐹‘(𝑓𝑦)) ↔ 𝑓 ∈ dom (( E ∘ Domain) ∖ Fix (Apply ∘ (FullFun𝐹 ∘ Restrict))))
6463con1bii 358 . . . . . . . 8 𝑓 ∈ dom (( E ∘ Domain) ∖ Fix (Apply ∘ (FullFun𝐹 ∘ Restrict))) ↔ ∀𝑦 ∈ dom 𝑓(𝑓𝑦) = (𝐹‘(𝑓𝑦)))
6514, 64anbi12i 626 . . . . . . 7 ((𝑓 ∈ ( Funs ∩ (Domain “ On)) ∧ ¬ 𝑓 ∈ dom (( E ∘ Domain) ∖ Fix (Apply ∘ (FullFun𝐹 ∘ Restrict)))) ↔ ((Fun 𝑓 ∧ dom 𝑓 ∈ On) ∧ ∀𝑦 ∈ dom 𝑓(𝑓𝑦) = (𝐹‘(𝑓𝑦))))
66 anass 469 . . . . . . 7 (((Fun 𝑓 ∧ dom 𝑓 ∈ On) ∧ ∀𝑦 ∈ dom 𝑓(𝑓𝑦) = (𝐹‘(𝑓𝑦))) ↔ (Fun 𝑓 ∧ (dom 𝑓 ∈ On ∧ ∀𝑦 ∈ dom 𝑓(𝑓𝑦) = (𝐹‘(𝑓𝑦)))))
6765, 66bitri 276 . . . . . 6 ((𝑓 ∈ ( Funs ∩ (Domain “ On)) ∧ ¬ 𝑓 ∈ dom (( E ∘ Domain) ∖ Fix (Apply ∘ (FullFun𝐹 ∘ Restrict)))) ↔ (Fun 𝑓 ∧ (dom 𝑓 ∈ On ∧ ∀𝑦 ∈ dom 𝑓(𝑓𝑦) = (𝐹‘(𝑓𝑦)))))
68 eleq1 2897 . . . . . . . . 9 (𝑥 = dom 𝑓 → (𝑥 ∈ On ↔ dom 𝑓 ∈ On))
69 raleq 3403 . . . . . . . . 9 (𝑥 = dom 𝑓 → (∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)) ↔ ∀𝑦 ∈ dom 𝑓(𝑓𝑦) = (𝐹‘(𝑓𝑦))))
7068, 69anbi12d 630 . . . . . . . 8 (𝑥 = dom 𝑓 → ((𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))) ↔ (dom 𝑓 ∈ On ∧ ∀𝑦 ∈ dom 𝑓(𝑓𝑦) = (𝐹‘(𝑓𝑦)))))
7170anbi2d 628 . . . . . . 7 (𝑥 = dom 𝑓 → ((Fun 𝑓 ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))) ↔ (Fun 𝑓 ∧ (dom 𝑓 ∈ On ∧ ∀𝑦 ∈ dom 𝑓(𝑓𝑦) = (𝐹‘(𝑓𝑦))))))
7221, 71ceqsexv 3539 . . . . . 6 (∃𝑥(𝑥 = dom 𝑓 ∧ (Fun 𝑓 ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))))) ↔ (Fun 𝑓 ∧ (dom 𝑓 ∈ On ∧ ∀𝑦 ∈ dom 𝑓(𝑓𝑦) = (𝐹‘(𝑓𝑦)))))
73 df-fn 6351 . . . . . . . . . 10 (𝑓 Fn 𝑥 ↔ (Fun 𝑓 ∧ dom 𝑓 = 𝑥))
74 eqcom 2825 . . . . . . . . . . 11 (dom 𝑓 = 𝑥𝑥 = dom 𝑓)
7574anbi2i 622 . . . . . . . . . 10 ((Fun 𝑓 ∧ dom 𝑓 = 𝑥) ↔ (Fun 𝑓𝑥 = dom 𝑓))
76 ancom 461 . . . . . . . . . 10 ((Fun 𝑓𝑥 = dom 𝑓) ↔ (𝑥 = dom 𝑓 ∧ Fun 𝑓))
7773, 75, 763bitri 298 . . . . . . . . 9 (𝑓 Fn 𝑥 ↔ (𝑥 = dom 𝑓 ∧ Fun 𝑓))
7877anbi1i 623 . . . . . . . 8 ((𝑓 Fn 𝑥 ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))) ↔ ((𝑥 = dom 𝑓 ∧ Fun 𝑓) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))))
79 an12 641 . . . . . . . 8 ((𝑓 Fn 𝑥 ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))) ↔ (𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))))
80 anass 469 . . . . . . . 8 (((𝑥 = dom 𝑓 ∧ Fun 𝑓) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))) ↔ (𝑥 = dom 𝑓 ∧ (Fun 𝑓 ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))))))
8178, 79, 803bitr3ri 303 . . . . . . 7 ((𝑥 = dom 𝑓 ∧ (Fun 𝑓 ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))))) ↔ (𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))))
8281exbii 1839 . . . . . 6 (∃𝑥(𝑥 = dom 𝑓 ∧ (Fun 𝑓 ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))))) ↔ ∃𝑥(𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))))
8367, 72, 823bitr2i 300 . . . . 5 ((𝑓 ∈ ( Funs ∩ (Domain “ On)) ∧ ¬ 𝑓 ∈ dom (( E ∘ Domain) ∖ Fix (Apply ∘ (FullFun𝐹 ∘ Restrict)))) ↔ ∃𝑥(𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))))
84 eldif 3943 . . . . 5 (𝑓 ∈ (( Funs ∩ (Domain “ On)) ∖ dom (( E ∘ Domain) ∖ Fix (Apply ∘ (FullFun𝐹 ∘ Restrict)))) ↔ (𝑓 ∈ ( Funs ∩ (Domain “ On)) ∧ ¬ 𝑓 ∈ dom (( E ∘ Domain) ∖ Fix (Apply ∘ (FullFun𝐹 ∘ Restrict)))))
85 df-rex 3141 . . . . 5 (∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))) ↔ ∃𝑥(𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))))
8683, 84, 853bitr4i 304 . . . 4 (𝑓 ∈ (( Funs ∩ (Domain “ On)) ∖ dom (( E ∘ Domain) ∖ Fix (Apply ∘ (FullFun𝐹 ∘ Restrict)))) ↔ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))))
8786abbi2i 2950 . . 3 (( Funs ∩ (Domain “ On)) ∖ dom (( E ∘ Domain) ∖ Fix (Apply ∘ (FullFun𝐹 ∘ Restrict)))) = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
8887unieqi 4839 . 2 (( Funs ∩ (Domain “ On)) ∖ dom (( E ∘ Domain) ∖ Fix (Apply ∘ (FullFun𝐹 ∘ Restrict)))) = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
891, 88eqtr4i 2844 1 recs(𝐹) = (( Funs ∩ (Domain “ On)) ∖ dom (( E ∘ Domain) ∖ Fix (Apply ∘ (FullFun𝐹 ∘ Restrict))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396   = wceq 1528  wex 1771  wcel 2105  {cab 2796  wral 3135  wrex 3136  cdif 3930  cin 3932  cop 4563   cuni 4830   class class class wbr 5057   E cep 5457  ccnv 5547  dom cdm 5548  cres 5550  cima 5551  ccom 5552  Oncon0 6184  Fun wfun 6342   Fn wfn 6343  cfv 6348  recscrecs 7996   Fix cfix 33193   Funs cfuns 33195  Domaincdomain 33201  Applycapply 33203  FullFuncfullfn 33208  Restrictcrestrict 33209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-symdif 4216  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fo 6354  df-fv 6356  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-txp 33212  df-pprod 33213  df-bigcup 33216  df-fix 33217  df-funs 33219  df-singleton 33220  df-singles 33221  df-image 33222  df-cart 33223  df-img 33224  df-domain 33225  df-range 33226  df-cap 33228  df-restrict 33229  df-apply 33231  df-funpart 33232  df-fullfun 33233
This theorem is referenced by: (None)
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