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Theorem fullfo 17966
Description: The morphism map of a full functor is a surjection. (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)
fullfo.f (𝜑𝐹(𝐶 Full 𝐷)𝐺)
fullfo.x (𝜑𝑋𝐵)
fullfo.y (𝜑𝑌𝐵)
Assertion
Ref Expression
fullfo (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–onto→((𝐹𝑋)𝐽(𝐹𝑌)))

Proof of Theorem fullfo
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fullfo.f . . 3 (𝜑𝐹(𝐶 Full 𝐷)𝐺)
2 isfull.b . . . . 5 𝐵 = (Base‘𝐶)
3 isfull.j . . . . 5 𝐽 = (Hom ‘𝐷)
4 isfull.h . . . . 5 𝐻 = (Hom ‘𝐶)
52, 3, 4isfull2 17965 . . . 4 (𝐹(𝐶 Full 𝐷)𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹𝑥)𝐽(𝐹𝑦))))
65simprbi 496 . . 3 (𝐹(𝐶 Full 𝐷)𝐺 → ∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹𝑥)𝐽(𝐹𝑦)))
71, 6syl 17 . 2 (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹𝑥)𝐽(𝐹𝑦)))
8 fullfo.x . . 3 (𝜑𝑋𝐵)
9 fullfo.y . . . . 5 (𝜑𝑌𝐵)
109adantr 480 . . . 4 ((𝜑𝑥 = 𝑋) → 𝑌𝐵)
11 simplr 769 . . . . . 6 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑥 = 𝑋)
12 simpr 484 . . . . . 6 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌)
1311, 12oveq12d 7449 . . . . 5 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → (𝑥𝐺𝑦) = (𝑋𝐺𝑌))
1411, 12oveq12d 7449 . . . . 5 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
1511fveq2d 6911 . . . . . 6 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → (𝐹𝑥) = (𝐹𝑋))
1612fveq2d 6911 . . . . . 6 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → (𝐹𝑦) = (𝐹𝑌))
1715, 16oveq12d 7449 . . . . 5 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → ((𝐹𝑥)𝐽(𝐹𝑦)) = ((𝐹𝑋)𝐽(𝐹𝑌)))
1813, 14, 17foeq123d 6842 . . . 4 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → ((𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹𝑥)𝐽(𝐹𝑦)) ↔ (𝑋𝐺𝑌):(𝑋𝐻𝑌)–onto→((𝐹𝑋)𝐽(𝐹𝑌))))
1910, 18rspcdv 3614 . . 3 ((𝜑𝑥 = 𝑋) → (∀𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹𝑥)𝐽(𝐹𝑦)) → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–onto→((𝐹𝑋)𝐽(𝐹𝑌))))
208, 19rspcimdv 3612 . 2 (𝜑 → (∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹𝑥)𝐽(𝐹𝑦)) → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–onto→((𝐹𝑋)𝐽(𝐹𝑌))))
217, 20mpd 15 1 (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–onto→((𝐹𝑋)𝐽(𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wral 3059   class class class wbr 5148  ontowfo 6561  cfv 6563  (class class class)co 7431  Basecbs 17245  Hom chom 17309   Func cfunc 17905   Full cful 17956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-ixp 8937  df-func 17909  df-full 17958
This theorem is referenced by:  fulli  17967  ffthf1o  17973  fulloppc  17976  cofull  17988
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