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Theorem isfull2 16394
Description: Equivalent condition for a full functor. (Contributed by Mario Carneiro, 27-Jan-2017.)
Hypotheses
Ref Expression
isfull.b 𝐵 = (Base‘𝐶)
isfull.j 𝐽 = (Hom ‘𝐷)
isfull.h 𝐻 = (Hom ‘𝐶)
Assertion
Ref Expression
isfull2 (𝐹(𝐶 Full 𝐷)𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹𝑥)𝐽(𝐹𝑦))))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦   𝑥,𝐻,𝑦   𝑥,𝐽,𝑦   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦

Proof of Theorem isfull2
StepHypRef Expression
1 isfull.b . . 3 𝐵 = (Base‘𝐶)
2 isfull.j . . 3 𝐽 = (Hom ‘𝐷)
31, 2isfull 16393 . 2 (𝐹(𝐶 Full 𝐷)𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥𝐵𝑦𝐵 ran (𝑥𝐺𝑦) = ((𝐹𝑥)𝐽(𝐹𝑦))))
4 isfull.h . . . . . . 7 𝐻 = (Hom ‘𝐶)
5 simpll 786 . . . . . . 7 (((𝐹(𝐶 Func 𝐷)𝐺𝑥𝐵) ∧ 𝑦𝐵) → 𝐹(𝐶 Func 𝐷)𝐺)
6 simplr 788 . . . . . . 7 (((𝐹(𝐶 Func 𝐷)𝐺𝑥𝐵) ∧ 𝑦𝐵) → 𝑥𝐵)
7 simpr 476 . . . . . . 7 (((𝐹(𝐶 Func 𝐷)𝐺𝑥𝐵) ∧ 𝑦𝐵) → 𝑦𝐵)
81, 4, 2, 5, 6, 7funcf2 16351 . . . . . 6 (((𝐹(𝐶 Func 𝐷)𝐺𝑥𝐵) ∧ 𝑦𝐵) → (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)𝐽(𝐹𝑦)))
9 ffn 5958 . . . . . 6 ((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)𝐽(𝐹𝑦)) → (𝑥𝐺𝑦) Fn (𝑥𝐻𝑦))
10 df-fo 5810 . . . . . . 7 ((𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹𝑥)𝐽(𝐹𝑦)) ↔ ((𝑥𝐺𝑦) Fn (𝑥𝐻𝑦) ∧ ran (𝑥𝐺𝑦) = ((𝐹𝑥)𝐽(𝐹𝑦))))
1110baib 942 . . . . . 6 ((𝑥𝐺𝑦) Fn (𝑥𝐻𝑦) → ((𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹𝑥)𝐽(𝐹𝑦)) ↔ ran (𝑥𝐺𝑦) = ((𝐹𝑥)𝐽(𝐹𝑦))))
128, 9, 113syl 18 . . . . 5 (((𝐹(𝐶 Func 𝐷)𝐺𝑥𝐵) ∧ 𝑦𝐵) → ((𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹𝑥)𝐽(𝐹𝑦)) ↔ ran (𝑥𝐺𝑦) = ((𝐹𝑥)𝐽(𝐹𝑦))))
1312ralbidva 2968 . . . 4 ((𝐹(𝐶 Func 𝐷)𝐺𝑥𝐵) → (∀𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹𝑥)𝐽(𝐹𝑦)) ↔ ∀𝑦𝐵 ran (𝑥𝐺𝑦) = ((𝐹𝑥)𝐽(𝐹𝑦))))
1413ralbidva 2968 . . 3 (𝐹(𝐶 Func 𝐷)𝐺 → (∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹𝑥)𝐽(𝐹𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 ran (𝑥𝐺𝑦) = ((𝐹𝑥)𝐽(𝐹𝑦))))
1514pm5.32i 667 . 2 ((𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹𝑥)𝐽(𝐹𝑦))) ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥𝐵𝑦𝐵 ran (𝑥𝐺𝑦) = ((𝐹𝑥)𝐽(𝐹𝑦))))
163, 15bitr4i 266 1 (𝐹(𝐶 Full 𝐷)𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹𝑥)𝐽(𝐹𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wb 195  wa 383   = wceq 1475  wcel 1977  wral 2896   class class class wbr 4583  ran crn 5039   Fn wfn 5799  wf 5800  ontowfo 5802  cfv 5804  (class class class)co 6549  Basecbs 15695  Hom chom 15779   Func cfunc 16337   Full cful 16385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-ixp 7795  df-func 16341  df-full 16387
This theorem is referenced by:  fullfo  16395  isffth2  16399  cofull  16417  fullestrcsetc  16614  fullsetcestrc  16629
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