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Theorem isffth2 17876
Description: A fully faithful functor is a functor which is bijective on hom-sets. (Contributed by Mario Carneiro, 27-Jan-2017.)
Hypotheses
Ref Expression
isfth.b 𝐵 = (Base‘𝐶)
isfth.h 𝐻 = (Hom ‘𝐶)
isfth.j 𝐽 = (Hom ‘𝐷)
Assertion
Ref Expression
isffth2 (𝐹((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷))𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1-onto→((𝐹𝑥)𝐽(𝐹𝑦))))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦   𝑥,𝐻,𝑦   𝑥,𝐽,𝑦

Proof of Theorem isffth2
StepHypRef Expression
1 isfth.b . . . 4 𝐵 = (Base‘𝐶)
2 isfth.j . . . 4 𝐽 = (Hom ‘𝐷)
3 isfth.h . . . 4 𝐻 = (Hom ‘𝐶)
41, 2, 3isfull2 17871 . . 3 (𝐹(𝐶 Full 𝐷)𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹𝑥)𝐽(𝐹𝑦))))
51, 3, 2isfth2 17875 . . 3 (𝐹(𝐶 Faith 𝐷)𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1→((𝐹𝑥)𝐽(𝐹𝑦))))
64, 5anbi12i 626 . 2 ((𝐹(𝐶 Full 𝐷)𝐺𝐹(𝐶 Faith 𝐷)𝐺) ↔ ((𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹𝑥)𝐽(𝐹𝑦))) ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1→((𝐹𝑥)𝐽(𝐹𝑦)))))
7 brin 5193 . 2 (𝐹((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷))𝐺 ↔ (𝐹(𝐶 Full 𝐷)𝐺𝐹(𝐶 Faith 𝐷)𝐺))
8 df-f1o 6543 . . . . . . 7 ((𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1-onto→((𝐹𝑥)𝐽(𝐹𝑦)) ↔ ((𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1→((𝐹𝑥)𝐽(𝐹𝑦)) ∧ (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹𝑥)𝐽(𝐹𝑦))))
98biancomi 462 . . . . . 6 ((𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1-onto→((𝐹𝑥)𝐽(𝐹𝑦)) ↔ ((𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹𝑥)𝐽(𝐹𝑦)) ∧ (𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1→((𝐹𝑥)𝐽(𝐹𝑦))))
1092ralbii 3122 . . . . 5 (∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1-onto→((𝐹𝑥)𝐽(𝐹𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 ((𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹𝑥)𝐽(𝐹𝑦)) ∧ (𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1→((𝐹𝑥)𝐽(𝐹𝑦))))
11 r19.26-2 3132 . . . . 5 (∀𝑥𝐵𝑦𝐵 ((𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹𝑥)𝐽(𝐹𝑦)) ∧ (𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1→((𝐹𝑥)𝐽(𝐹𝑦))) ↔ (∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹𝑥)𝐽(𝐹𝑦)) ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1→((𝐹𝑥)𝐽(𝐹𝑦))))
1210, 11bitri 275 . . . 4 (∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1-onto→((𝐹𝑥)𝐽(𝐹𝑦)) ↔ (∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹𝑥)𝐽(𝐹𝑦)) ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1→((𝐹𝑥)𝐽(𝐹𝑦))))
1312anbi2i 622 . . 3 ((𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1-onto→((𝐹𝑥)𝐽(𝐹𝑦))) ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ (∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹𝑥)𝐽(𝐹𝑦)) ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1→((𝐹𝑥)𝐽(𝐹𝑦)))))
14 anandi 673 . . 3 ((𝐹(𝐶 Func 𝐷)𝐺 ∧ (∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹𝑥)𝐽(𝐹𝑦)) ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1→((𝐹𝑥)𝐽(𝐹𝑦)))) ↔ ((𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹𝑥)𝐽(𝐹𝑦))) ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1→((𝐹𝑥)𝐽(𝐹𝑦)))))
1513, 14bitri 275 . 2 ((𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1-onto→((𝐹𝑥)𝐽(𝐹𝑦))) ↔ ((𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹𝑥)𝐽(𝐹𝑦))) ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1→((𝐹𝑥)𝐽(𝐹𝑦)))))
166, 7, 153bitr4i 303 1 (𝐹((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷))𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1-onto→((𝐹𝑥)𝐽(𝐹𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1533  wral 3055  cin 3942   class class class wbr 5141  1-1wf1 6533  ontowfo 6534  1-1-ontowf1o 6535  cfv 6536  (class class class)co 7404  Basecbs 17151  Hom chom 17215   Func cfunc 17811   Full cful 17862   Faith cfth 17863
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7971  df-2nd 7972  df-map 8821  df-ixp 8891  df-func 17815  df-full 17864  df-fth 17865
This theorem is referenced by:  idffth  17893  ressffth  17898  catciso  18071  yonffthlem  18245
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