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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  –onto→wfo 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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