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Theorem fulli 16396
 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)
fulli.r (𝜑𝑅 ∈ ((𝐹𝑋)𝐽(𝐹𝑌)))
Assertion
Ref Expression
fulli (𝜑 → ∃𝑓 ∈ (𝑋𝐻𝑌)𝑅 = ((𝑋𝐺𝑌)‘𝑓))
Distinct variable groups:   𝐵,𝑓   𝐶,𝑓   𝐷,𝑓   𝑓,𝐻   𝑓,𝐽   𝑅,𝑓   𝑓,𝑋   𝑓,𝑌   𝑓,𝐹   𝑓,𝐺
Allowed substitution hint:   𝜑(𝑓)

Proof of Theorem fulli
StepHypRef Expression
1 isfull.b . . 3 𝐵 = (Base‘𝐶)
2 isfull.j . . 3 𝐽 = (Hom ‘𝐷)
3 isfull.h . . 3 𝐻 = (Hom ‘𝐶)
4 fullfo.f . . 3 (𝜑𝐹(𝐶 Full 𝐷)𝐺)
5 fullfo.x . . 3 (𝜑𝑋𝐵)
6 fullfo.y . . 3 (𝜑𝑌𝐵)
71, 2, 3, 4, 5, 6fullfo 16395 . 2 (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–onto→((𝐹𝑋)𝐽(𝐹𝑌)))
8 fulli.r . 2 (𝜑𝑅 ∈ ((𝐹𝑋)𝐽(𝐹𝑌)))
9 foelrn 6286 . 2 (((𝑋𝐺𝑌):(𝑋𝐻𝑌)–onto→((𝐹𝑋)𝐽(𝐹𝑌)) ∧ 𝑅 ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) → ∃𝑓 ∈ (𝑋𝐻𝑌)𝑅 = ((𝑋𝐺𝑌)‘𝑓))
107, 8, 9syl2anc 691 1 (𝜑 → ∃𝑓 ∈ (𝑋𝐻𝑌)𝑅 = ((𝑋𝐺𝑌)‘𝑓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ∃wrex 2897   class class class wbr 4583  –onto→wfo 5802  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  Hom chom 15779   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:  ffthiso  16412
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