MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isfull2 Structured version   Visualization version   GIF version

Theorem isfull2 17857
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 17856 . 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 766 . . . . . . 7 (((𝐹(𝐶 Func 𝐷)𝐺𝑥𝐵) ∧ 𝑦𝐵) → 𝐹(𝐶 Func 𝐷)𝐺)
6 simplr 768 . . . . . . 7 (((𝐹(𝐶 Func 𝐷)𝐺𝑥𝐵) ∧ 𝑦𝐵) → 𝑥𝐵)
7 simpr 486 . . . . . . 7 (((𝐹(𝐶 Func 𝐷)𝐺𝑥𝐵) ∧ 𝑦𝐵) → 𝑦𝐵)
81, 4, 2, 5, 6, 7funcf2 17813 . . . . . 6 (((𝐹(𝐶 Func 𝐷)𝐺𝑥𝐵) ∧ 𝑦𝐵) → (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)𝐽(𝐹𝑦)))
9 ffn 6713 . . . . . 6 ((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)𝐽(𝐹𝑦)) → (𝑥𝐺𝑦) Fn (𝑥𝐻𝑦))
10 df-fo 6545 . . . . . . 7 ((𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹𝑥)𝐽(𝐹𝑦)) ↔ ((𝑥𝐺𝑦) Fn (𝑥𝐻𝑦) ∧ ran (𝑥𝐺𝑦) = ((𝐹𝑥)𝐽(𝐹𝑦))))
1110baib 537 . . . . . 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 3176 . . . 4 ((𝐹(𝐶 Func 𝐷)𝐺𝑥𝐵) → (∀𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹𝑥)𝐽(𝐹𝑦)) ↔ ∀𝑦𝐵 ran (𝑥𝐺𝑦) = ((𝐹𝑥)𝐽(𝐹𝑦))))
1413ralbidva 3176 . . 3 (𝐹(𝐶 Func 𝐷)𝐺 → (∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹𝑥)𝐽(𝐹𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 ran (𝑥𝐺𝑦) = ((𝐹𝑥)𝐽(𝐹𝑦))))
1514pm5.32i 576 . 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 278 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 205  wa 397   = wceq 1542  wcel 2107  wral 3062   class class class wbr 5146  ran crn 5675   Fn wfn 6534  wf 6535  ontowfo 6537  cfv 6539  (class class class)co 7403  Basecbs 17139  Hom chom 17203   Func cfunc 17799   Full cful 17848
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-rep 5283  ax-sep 5297  ax-nul 5304  ax-pow 5361  ax-pr 5425  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  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-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4527  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4907  df-iun 4997  df-br 5147  df-opab 5209  df-mpt 5230  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6491  df-fun 6541  df-fn 6542  df-f 6543  df-fo 6545  df-fv 6547  df-ov 7406  df-oprab 7407  df-mpo 7408  df-1st 7969  df-2nd 7970  df-map 8817  df-ixp 8887  df-func 17803  df-full 17850
This theorem is referenced by:  fullfo  17858  isffth2  17862  cofull  17880  fullestrcsetc  18098  fullsetcestrc  18113  fullthinc  47567
  Copyright terms: Public domain W3C validator