Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dfrecs2 Structured version   Visualization version   GIF version

Theorem dfrecs2 34922
Description: A quantifier-free definition of recs. (Contributed by Scott Fenton, 17-Jul-2020.)
Assertion
Ref Expression
dfrecs2 recs(𝐹) = βˆͺ (( Funs ∩ (β—‘Domain β€œ On)) βˆ– dom ((β—‘ E ∘ Domain) βˆ– Fix (β—‘Apply ∘ (FullFun𝐹 ∘ Restrict))))

Proof of Theorem dfrecs2
Dummy variables 𝑓 π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfrecs3 8372 . 2 recs(𝐹) = βˆͺ {𝑓 ∣ βˆƒπ‘₯ ∈ On (𝑓 Fn π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦)))}
2 elin 3965 . . . . . . . . 9 (𝑓 ∈ ( Funs ∩ (β—‘Domain β€œ On)) ↔ (𝑓 ∈ Funs ∧ 𝑓 ∈ (β—‘Domain β€œ On)))
3 vex 3479 . . . . . . . . . . 11 𝑓 ∈ V
43elfuns 34887 . . . . . . . . . 10 (𝑓 ∈ Funs ↔ Fun 𝑓)
5 vex 3479 . . . . . . . . . . . . . 14 π‘₯ ∈ V
65, 3brcnv 5883 . . . . . . . . . . . . 13 (π‘₯β—‘Domain𝑓 ↔ 𝑓Domainπ‘₯)
73, 5brdomain 34905 . . . . . . . . . . . . 13 (𝑓Domainπ‘₯ ↔ π‘₯ = dom 𝑓)
86, 7bitri 275 . . . . . . . . . . . 12 (π‘₯β—‘Domain𝑓 ↔ π‘₯ = dom 𝑓)
98rexbii 3095 . . . . . . . . . . 11 (βˆƒπ‘₯ ∈ On π‘₯β—‘Domain𝑓 ↔ βˆƒπ‘₯ ∈ On π‘₯ = dom 𝑓)
103elima 6065 . . . . . . . . . . 11 (𝑓 ∈ (β—‘Domain β€œ On) ↔ βˆƒπ‘₯ ∈ On π‘₯β—‘Domain𝑓)
11 risset 3231 . . . . . . . . . . 11 (dom 𝑓 ∈ On ↔ βˆƒπ‘₯ ∈ On π‘₯ = dom 𝑓)
129, 10, 113bitr4i 303 . . . . . . . . . 10 (𝑓 ∈ (β—‘Domain β€œ On) ↔ dom 𝑓 ∈ On)
134, 12anbi12i 628 . . . . . . . . 9 ((𝑓 ∈ Funs ∧ 𝑓 ∈ (β—‘Domain β€œ On)) ↔ (Fun 𝑓 ∧ dom 𝑓 ∈ On))
142, 13bitri 275 . . . . . . . 8 (𝑓 ∈ ( Funs ∩ (β—‘Domain β€œ On)) ↔ (Fun 𝑓 ∧ dom 𝑓 ∈ On))
153eldm 5901 . . . . . . . . . . 11 (𝑓 ∈ dom ((β—‘ E ∘ Domain) βˆ– Fix (β—‘Apply ∘ (FullFun𝐹 ∘ Restrict))) ↔ βˆƒπ‘¦ 𝑓((β—‘ E ∘ Domain) βˆ– Fix (β—‘Apply ∘ (FullFun𝐹 ∘ Restrict)))𝑦)
16 brdif 5202 . . . . . . . . . . . . 13 (𝑓((β—‘ E ∘ Domain) βˆ– Fix (β—‘Apply ∘ (FullFun𝐹 ∘ Restrict)))𝑦 ↔ (𝑓(β—‘ E ∘ Domain)𝑦 ∧ Β¬ 𝑓 Fix (β—‘Apply ∘ (FullFun𝐹 ∘ Restrict))𝑦))
17 vex 3479 . . . . . . . . . . . . . . . 16 𝑦 ∈ V
183, 17brco 5871 . . . . . . . . . . . . . . 15 (𝑓(β—‘ E ∘ Domain)𝑦 ↔ βˆƒπ‘₯(𝑓Domainπ‘₯ ∧ π‘₯β—‘ E 𝑦))
197anbi1i 625 . . . . . . . . . . . . . . . . 17 ((𝑓Domainπ‘₯ ∧ π‘₯β—‘ E 𝑦) ↔ (π‘₯ = dom 𝑓 ∧ π‘₯β—‘ E 𝑦))
2019exbii 1851 . . . . . . . . . . . . . . . 16 (βˆƒπ‘₯(𝑓Domainπ‘₯ ∧ π‘₯β—‘ E 𝑦) ↔ βˆƒπ‘₯(π‘₯ = dom 𝑓 ∧ π‘₯β—‘ E 𝑦))
213dmex 7902 . . . . . . . . . . . . . . . . 17 dom 𝑓 ∈ V
22 breq1 5152 . . . . . . . . . . . . . . . . 17 (π‘₯ = dom 𝑓 β†’ (π‘₯β—‘ E 𝑦 ↔ dom 𝑓◑ E 𝑦))
2321, 22ceqsexv 3526 . . . . . . . . . . . . . . . 16 (βˆƒπ‘₯(π‘₯ = dom 𝑓 ∧ π‘₯β—‘ E 𝑦) ↔ dom 𝑓◑ E 𝑦)
2420, 23bitri 275 . . . . . . . . . . . . . . 15 (βˆƒπ‘₯(𝑓Domainπ‘₯ ∧ π‘₯β—‘ E 𝑦) ↔ dom 𝑓◑ E 𝑦)
2521, 17brcnv 5883 . . . . . . . . . . . . . . . 16 (dom 𝑓◑ E 𝑦 ↔ 𝑦 E dom 𝑓)
2621epeli 5583 . . . . . . . . . . . . . . . 16 (𝑦 E dom 𝑓 ↔ 𝑦 ∈ dom 𝑓)
2725, 26bitri 275 . . . . . . . . . . . . . . 15 (dom 𝑓◑ E 𝑦 ↔ 𝑦 ∈ dom 𝑓)
2818, 24, 273bitri 297 . . . . . . . . . . . . . 14 (𝑓(β—‘ E ∘ Domain)𝑦 ↔ 𝑦 ∈ dom 𝑓)
29 df-br 5150 . . . . . . . . . . . . . . . 16 (𝑓 Fix (β—‘Apply ∘ (FullFun𝐹 ∘ Restrict))𝑦 ↔ βŸ¨π‘“, π‘¦βŸ© ∈ Fix (β—‘Apply ∘ (FullFun𝐹 ∘ Restrict)))
30 opex 5465 . . . . . . . . . . . . . . . . 17 βŸ¨π‘“, π‘¦βŸ© ∈ V
3130elfix 34875 . . . . . . . . . . . . . . . 16 (βŸ¨π‘“, π‘¦βŸ© ∈ Fix (β—‘Apply ∘ (FullFun𝐹 ∘ Restrict)) ↔ βŸ¨π‘“, π‘¦βŸ©(β—‘Apply ∘ (FullFun𝐹 ∘ Restrict))βŸ¨π‘“, π‘¦βŸ©)
3230, 30brco 5871 . . . . . . . . . . . . . . . . 17 (βŸ¨π‘“, π‘¦βŸ©(β—‘Apply ∘ (FullFun𝐹 ∘ Restrict))βŸ¨π‘“, π‘¦βŸ© ↔ βˆƒπ‘₯(βŸ¨π‘“, π‘¦βŸ©(FullFun𝐹 ∘ Restrict)π‘₯ ∧ π‘₯β—‘ApplyβŸ¨π‘“, π‘¦βŸ©))
33 ancom 462 . . . . . . . . . . . . . . . . . . . 20 ((βŸ¨π‘“, π‘¦βŸ©(FullFun𝐹 ∘ Restrict)π‘₯ ∧ π‘₯β—‘ApplyβŸ¨π‘“, π‘¦βŸ©) ↔ (π‘₯β—‘ApplyβŸ¨π‘“, π‘¦βŸ© ∧ βŸ¨π‘“, π‘¦βŸ©(FullFun𝐹 ∘ Restrict)π‘₯))
345, 30brcnv 5883 . . . . . . . . . . . . . . . . . . . . . 22 (π‘₯β—‘ApplyβŸ¨π‘“, π‘¦βŸ© ↔ βŸ¨π‘“, π‘¦βŸ©Applyπ‘₯)
353, 17, 5brapply 34910 . . . . . . . . . . . . . . . . . . . . . 22 (βŸ¨π‘“, π‘¦βŸ©Applyπ‘₯ ↔ π‘₯ = (π‘“β€˜π‘¦))
3634, 35bitri 275 . . . . . . . . . . . . . . . . . . . . 21 (π‘₯β—‘ApplyβŸ¨π‘“, π‘¦βŸ© ↔ π‘₯ = (π‘“β€˜π‘¦))
3736anbi1i 625 . . . . . . . . . . . . . . . . . . . 20 ((π‘₯β—‘ApplyβŸ¨π‘“, π‘¦βŸ© ∧ βŸ¨π‘“, π‘¦βŸ©(FullFun𝐹 ∘ Restrict)π‘₯) ↔ (π‘₯ = (π‘“β€˜π‘¦) ∧ βŸ¨π‘“, π‘¦βŸ©(FullFun𝐹 ∘ Restrict)π‘₯))
3833, 37bitri 275 . . . . . . . . . . . . . . . . . . 19 ((βŸ¨π‘“, π‘¦βŸ©(FullFun𝐹 ∘ Restrict)π‘₯ ∧ π‘₯β—‘ApplyβŸ¨π‘“, π‘¦βŸ©) ↔ (π‘₯ = (π‘“β€˜π‘¦) ∧ βŸ¨π‘“, π‘¦βŸ©(FullFun𝐹 ∘ Restrict)π‘₯))
3938exbii 1851 . . . . . . . . . . . . . . . . . 18 (βˆƒπ‘₯(βŸ¨π‘“, π‘¦βŸ©(FullFun𝐹 ∘ Restrict)π‘₯ ∧ π‘₯β—‘ApplyβŸ¨π‘“, π‘¦βŸ©) ↔ βˆƒπ‘₯(π‘₯ = (π‘“β€˜π‘¦) ∧ βŸ¨π‘“, π‘¦βŸ©(FullFun𝐹 ∘ Restrict)π‘₯))
40 fvex 6905 . . . . . . . . . . . . . . . . . . 19 (π‘“β€˜π‘¦) ∈ V
41 breq2 5153 . . . . . . . . . . . . . . . . . . 19 (π‘₯ = (π‘“β€˜π‘¦) β†’ (βŸ¨π‘“, π‘¦βŸ©(FullFun𝐹 ∘ Restrict)π‘₯ ↔ βŸ¨π‘“, π‘¦βŸ©(FullFun𝐹 ∘ Restrict)(π‘“β€˜π‘¦)))
4240, 41ceqsexv 3526 . . . . . . . . . . . . . . . . . 18 (βˆƒπ‘₯(π‘₯ = (π‘“β€˜π‘¦) ∧ βŸ¨π‘“, π‘¦βŸ©(FullFun𝐹 ∘ Restrict)π‘₯) ↔ βŸ¨π‘“, π‘¦βŸ©(FullFun𝐹 ∘ Restrict)(π‘“β€˜π‘¦))
4339, 42bitri 275 . . . . . . . . . . . . . . . . 17 (βˆƒπ‘₯(βŸ¨π‘“, π‘¦βŸ©(FullFun𝐹 ∘ Restrict)π‘₯ ∧ π‘₯β—‘ApplyβŸ¨π‘“, π‘¦βŸ©) ↔ βŸ¨π‘“, π‘¦βŸ©(FullFun𝐹 ∘ Restrict)(π‘“β€˜π‘¦))
4430, 40brco 5871 . . . . . . . . . . . . . . . . . 18 (βŸ¨π‘“, π‘¦βŸ©(FullFun𝐹 ∘ Restrict)(π‘“β€˜π‘¦) ↔ βˆƒπ‘₯(βŸ¨π‘“, π‘¦βŸ©Restrictπ‘₯ ∧ π‘₯FullFun𝐹(π‘“β€˜π‘¦)))
453, 17, 5brrestrict 34921 . . . . . . . . . . . . . . . . . . . . 21 (βŸ¨π‘“, π‘¦βŸ©Restrictπ‘₯ ↔ π‘₯ = (𝑓 β†Ύ 𝑦))
4645anbi1i 625 . . . . . . . . . . . . . . . . . . . 20 ((βŸ¨π‘“, π‘¦βŸ©Restrictπ‘₯ ∧ π‘₯FullFun𝐹(π‘“β€˜π‘¦)) ↔ (π‘₯ = (𝑓 β†Ύ 𝑦) ∧ π‘₯FullFun𝐹(π‘“β€˜π‘¦)))
4746exbii 1851 . . . . . . . . . . . . . . . . . . 19 (βˆƒπ‘₯(βŸ¨π‘“, π‘¦βŸ©Restrictπ‘₯ ∧ π‘₯FullFun𝐹(π‘“β€˜π‘¦)) ↔ βˆƒπ‘₯(π‘₯ = (𝑓 β†Ύ 𝑦) ∧ π‘₯FullFun𝐹(π‘“β€˜π‘¦)))
483resex 6030 . . . . . . . . . . . . . . . . . . . 20 (𝑓 β†Ύ 𝑦) ∈ V
49 breq1 5152 . . . . . . . . . . . . . . . . . . . 20 (π‘₯ = (𝑓 β†Ύ 𝑦) β†’ (π‘₯FullFun𝐹(π‘“β€˜π‘¦) ↔ (𝑓 β†Ύ 𝑦)FullFun𝐹(π‘“β€˜π‘¦)))
5048, 49ceqsexv 3526 . . . . . . . . . . . . . . . . . . 19 (βˆƒπ‘₯(π‘₯ = (𝑓 β†Ύ 𝑦) ∧ π‘₯FullFun𝐹(π‘“β€˜π‘¦)) ↔ (𝑓 β†Ύ 𝑦)FullFun𝐹(π‘“β€˜π‘¦))
5147, 50bitri 275 . . . . . . . . . . . . . . . . . 18 (βˆƒπ‘₯(βŸ¨π‘“, π‘¦βŸ©Restrictπ‘₯ ∧ π‘₯FullFun𝐹(π‘“β€˜π‘¦)) ↔ (𝑓 β†Ύ 𝑦)FullFun𝐹(π‘“β€˜π‘¦))
5248, 40brfullfun 34920 . . . . . . . . . . . . . . . . . 18 ((𝑓 β†Ύ 𝑦)FullFun𝐹(π‘“β€˜π‘¦) ↔ (π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦)))
5344, 51, 523bitri 297 . . . . . . . . . . . . . . . . 17 (βŸ¨π‘“, π‘¦βŸ©(FullFun𝐹 ∘ Restrict)(π‘“β€˜π‘¦) ↔ (π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦)))
5432, 43, 533bitri 297 . . . . . . . . . . . . . . . 16 (βŸ¨π‘“, π‘¦βŸ©(β—‘Apply ∘ (FullFun𝐹 ∘ Restrict))βŸ¨π‘“, π‘¦βŸ© ↔ (π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦)))
5529, 31, 543bitri 297 . . . . . . . . . . . . . . 15 (𝑓 Fix (β—‘Apply ∘ (FullFun𝐹 ∘ Restrict))𝑦 ↔ (π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦)))
5655notbii 320 . . . . . . . . . . . . . 14 (Β¬ 𝑓 Fix (β—‘Apply ∘ (FullFun𝐹 ∘ Restrict))𝑦 ↔ Β¬ (π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦)))
5728, 56anbi12i 628 . . . . . . . . . . . . 13 ((𝑓(β—‘ E ∘ Domain)𝑦 ∧ Β¬ 𝑓 Fix (β—‘Apply ∘ (FullFun𝐹 ∘ Restrict))𝑦) ↔ (𝑦 ∈ dom 𝑓 ∧ Β¬ (π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦))))
5816, 57bitri 275 . . . . . . . . . . . 12 (𝑓((β—‘ E ∘ Domain) βˆ– Fix (β—‘Apply ∘ (FullFun𝐹 ∘ Restrict)))𝑦 ↔ (𝑦 ∈ dom 𝑓 ∧ Β¬ (π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦))))
5958exbii 1851 . . . . . . . . . . 11 (βˆƒπ‘¦ 𝑓((β—‘ E ∘ Domain) βˆ– Fix (β—‘Apply ∘ (FullFun𝐹 ∘ Restrict)))𝑦 ↔ βˆƒπ‘¦(𝑦 ∈ dom 𝑓 ∧ Β¬ (π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦))))
6015, 59bitri 275 . . . . . . . . . 10 (𝑓 ∈ dom ((β—‘ E ∘ Domain) βˆ– Fix (β—‘Apply ∘ (FullFun𝐹 ∘ Restrict))) ↔ βˆƒπ‘¦(𝑦 ∈ dom 𝑓 ∧ Β¬ (π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦))))
61 df-rex 3072 . . . . . . . . . 10 (βˆƒπ‘¦ ∈ dom 𝑓 Β¬ (π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦)) ↔ βˆƒπ‘¦(𝑦 ∈ dom 𝑓 ∧ Β¬ (π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦))))
62 rexnal 3101 . . . . . . . . . 10 (βˆƒπ‘¦ ∈ dom 𝑓 Β¬ (π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦)) ↔ Β¬ βˆ€π‘¦ ∈ dom 𝑓(π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦)))
6360, 61, 623bitr2ri 300 . . . . . . . . 9 (Β¬ βˆ€π‘¦ ∈ dom 𝑓(π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦)) ↔ 𝑓 ∈ dom ((β—‘ E ∘ Domain) βˆ– Fix (β—‘Apply ∘ (FullFun𝐹 ∘ Restrict))))
6463con1bii 357 . . . . . . . 8 (Β¬ 𝑓 ∈ dom ((β—‘ E ∘ Domain) βˆ– Fix (β—‘Apply ∘ (FullFun𝐹 ∘ Restrict))) ↔ βˆ€π‘¦ ∈ dom 𝑓(π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦)))
6514, 64anbi12i 628 . . . . . . 7 ((𝑓 ∈ ( Funs ∩ (β—‘Domain β€œ On)) ∧ Β¬ 𝑓 ∈ dom ((β—‘ E ∘ Domain) βˆ– Fix (β—‘Apply ∘ (FullFun𝐹 ∘ Restrict)))) ↔ ((Fun 𝑓 ∧ dom 𝑓 ∈ On) ∧ βˆ€π‘¦ ∈ dom 𝑓(π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦))))
66 anass 470 . . . . . . 7 (((Fun 𝑓 ∧ dom 𝑓 ∈ On) ∧ βˆ€π‘¦ ∈ dom 𝑓(π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦))) ↔ (Fun 𝑓 ∧ (dom 𝑓 ∈ On ∧ βˆ€π‘¦ ∈ dom 𝑓(π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦)))))
6765, 66bitri 275 . . . . . 6 ((𝑓 ∈ ( Funs ∩ (β—‘Domain β€œ On)) ∧ Β¬ 𝑓 ∈ dom ((β—‘ E ∘ Domain) βˆ– Fix (β—‘Apply ∘ (FullFun𝐹 ∘ Restrict)))) ↔ (Fun 𝑓 ∧ (dom 𝑓 ∈ On ∧ βˆ€π‘¦ ∈ dom 𝑓(π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦)))))
68 eleq1 2822 . . . . . . . . 9 (π‘₯ = dom 𝑓 β†’ (π‘₯ ∈ On ↔ dom 𝑓 ∈ On))
69 raleq 3323 . . . . . . . . 9 (π‘₯ = dom 𝑓 β†’ (βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦)) ↔ βˆ€π‘¦ ∈ dom 𝑓(π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦))))
7068, 69anbi12d 632 . . . . . . . 8 (π‘₯ = dom 𝑓 β†’ ((π‘₯ ∈ On ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦))) ↔ (dom 𝑓 ∈ On ∧ βˆ€π‘¦ ∈ dom 𝑓(π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦)))))
7170anbi2d 630 . . . . . . 7 (π‘₯ = dom 𝑓 β†’ ((Fun 𝑓 ∧ (π‘₯ ∈ On ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦)))) ↔ (Fun 𝑓 ∧ (dom 𝑓 ∈ On ∧ βˆ€π‘¦ ∈ dom 𝑓(π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦))))))
7221, 71ceqsexv 3526 . . . . . 6 (βˆƒπ‘₯(π‘₯ = dom 𝑓 ∧ (Fun 𝑓 ∧ (π‘₯ ∈ On ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦))))) ↔ (Fun 𝑓 ∧ (dom 𝑓 ∈ On ∧ βˆ€π‘¦ ∈ dom 𝑓(π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦)))))
73 df-fn 6547 . . . . . . . . . 10 (𝑓 Fn π‘₯ ↔ (Fun 𝑓 ∧ dom 𝑓 = π‘₯))
74 eqcom 2740 . . . . . . . . . . 11 (dom 𝑓 = π‘₯ ↔ π‘₯ = dom 𝑓)
7574anbi2i 624 . . . . . . . . . 10 ((Fun 𝑓 ∧ dom 𝑓 = π‘₯) ↔ (Fun 𝑓 ∧ π‘₯ = dom 𝑓))
76 ancom 462 . . . . . . . . . 10 ((Fun 𝑓 ∧ π‘₯ = dom 𝑓) ↔ (π‘₯ = dom 𝑓 ∧ Fun 𝑓))
7773, 75, 763bitri 297 . . . . . . . . 9 (𝑓 Fn π‘₯ ↔ (π‘₯ = dom 𝑓 ∧ Fun 𝑓))
7877anbi1i 625 . . . . . . . 8 ((𝑓 Fn π‘₯ ∧ (π‘₯ ∈ On ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦)))) ↔ ((π‘₯ = dom 𝑓 ∧ Fun 𝑓) ∧ (π‘₯ ∈ On ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦)))))
79 an12 644 . . . . . . . 8 ((𝑓 Fn π‘₯ ∧ (π‘₯ ∈ On ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦)))) ↔ (π‘₯ ∈ On ∧ (𝑓 Fn π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦)))))
80 anass 470 . . . . . . . 8 (((π‘₯ = dom 𝑓 ∧ Fun 𝑓) ∧ (π‘₯ ∈ On ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦)))) ↔ (π‘₯ = dom 𝑓 ∧ (Fun 𝑓 ∧ (π‘₯ ∈ On ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦))))))
8178, 79, 803bitr3ri 302 . . . . . . 7 ((π‘₯ = dom 𝑓 ∧ (Fun 𝑓 ∧ (π‘₯ ∈ On ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦))))) ↔ (π‘₯ ∈ On ∧ (𝑓 Fn π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦)))))
8281exbii 1851 . . . . . 6 (βˆƒπ‘₯(π‘₯ = dom 𝑓 ∧ (Fun 𝑓 ∧ (π‘₯ ∈ On ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦))))) ↔ βˆƒπ‘₯(π‘₯ ∈ On ∧ (𝑓 Fn π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦)))))
8367, 72, 823bitr2i 299 . . . . 5 ((𝑓 ∈ ( Funs ∩ (β—‘Domain β€œ On)) ∧ Β¬ 𝑓 ∈ dom ((β—‘ E ∘ Domain) βˆ– Fix (β—‘Apply ∘ (FullFun𝐹 ∘ Restrict)))) ↔ βˆƒπ‘₯(π‘₯ ∈ On ∧ (𝑓 Fn π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦)))))
84 eldif 3959 . . . . 5 (𝑓 ∈ (( Funs ∩ (β—‘Domain β€œ On)) βˆ– dom ((β—‘ E ∘ Domain) βˆ– Fix (β—‘Apply ∘ (FullFun𝐹 ∘ Restrict)))) ↔ (𝑓 ∈ ( Funs ∩ (β—‘Domain β€œ On)) ∧ Β¬ 𝑓 ∈ dom ((β—‘ E ∘ Domain) βˆ– Fix (β—‘Apply ∘ (FullFun𝐹 ∘ Restrict)))))
85 df-rex 3072 . . . . 5 (βˆƒπ‘₯ ∈ On (𝑓 Fn π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦))) ↔ βˆƒπ‘₯(π‘₯ ∈ On ∧ (𝑓 Fn π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦)))))
8683, 84, 853bitr4i 303 . . . 4 (𝑓 ∈ (( Funs ∩ (β—‘Domain β€œ On)) βˆ– dom ((β—‘ E ∘ Domain) βˆ– Fix (β—‘Apply ∘ (FullFun𝐹 ∘ Restrict)))) ↔ βˆƒπ‘₯ ∈ On (𝑓 Fn π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦))))
8786eqabi 2870 . . 3 (( Funs ∩ (β—‘Domain β€œ On)) βˆ– dom ((β—‘ E ∘ Domain) βˆ– Fix (β—‘Apply ∘ (FullFun𝐹 ∘ Restrict)))) = {𝑓 ∣ βˆƒπ‘₯ ∈ On (𝑓 Fn π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦)))}
8887unieqi 4922 . 2 βˆͺ (( Funs ∩ (β—‘Domain β€œ On)) βˆ– dom ((β—‘ E ∘ Domain) βˆ– Fix (β—‘Apply ∘ (FullFun𝐹 ∘ Restrict)))) = βˆͺ {𝑓 ∣ βˆƒπ‘₯ ∈ On (𝑓 Fn π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦)))}
891, 88eqtr4i 2764 1 recs(𝐹) = βˆͺ (( Funs ∩ (β—‘Domain β€œ On)) βˆ– dom ((β—‘ E ∘ Domain) βˆ– Fix (β—‘Apply ∘ (FullFun𝐹 ∘ Restrict))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   ∧ wa 397   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  {cab 2710  βˆ€wral 3062  βˆƒwrex 3071   βˆ– cdif 3946   ∩ cin 3948  βŸ¨cop 4635  βˆͺ cuni 4909   class class class wbr 5149   E cep 5580  β—‘ccnv 5676  dom cdm 5677   β†Ύ cres 5679   β€œ cima 5680   ∘ ccom 5681  Oncon0 6365  Fun wfun 6538   Fn wfn 6539  β€˜cfv 6544  recscrecs 8370   Fix cfix 34807   Funs cfuns 34809  Domaincdomain 34815  Applycapply 34817  FullFuncfullfn 34822  Restrictcrestrict 34823
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-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-symdif 4243  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550  df-fv 6552  df-ov 7412  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-txp 34826  df-pprod 34827  df-bigcup 34830  df-fix 34831  df-funs 34833  df-singleton 34834  df-singles 34835  df-image 34836  df-cart 34837  df-img 34838  df-domain 34839  df-range 34840  df-cap 34842  df-restrict 34843  df-apply 34845  df-funpart 34846  df-fullfun 34847
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator