Theorem fulli 17629

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	⊢ (𝜑 → 𝐹(𝐶 Full 𝐷)𝐺)
fullfo.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
fullfo.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
fulli.r	⊢ (𝜑 → 𝑅 ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))

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

Step	Hyp	Ref	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 ⊢ (𝜑 → 𝐹(𝐶 Full 𝐷)𝐺)
5		fullfo.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
6		fullfo.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
7	1, 2, 3, 4, 5, 6	fullfo 17628	. 2 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–onto→((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
8		fulli.r	. 2 ⊢ (𝜑 → 𝑅 ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
9		foelrn 6982	. 2 ⊢ (((𝑋𝐺𝑌):(𝑋𝐻𝑌)–onto→((𝐹‘𝑋)𝐽(𝐹‘𝑌)) ∧ 𝑅 ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌))) → ∃𝑓 ∈ (𝑋𝐻𝑌)𝑅 = ((𝑋𝐺𝑌)‘𝑓))
10	7, 8, 9	syl2anc 584	1 ⊢ (𝜑 → ∃𝑓 ∈ (𝑋𝐻𝑌)𝑅 = ((𝑋𝐺𝑌)‘𝑓))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  ∃wrex 3065   class class class wbr 5074  –onto→wfo 6431  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  Hom chom 16973   Full cful 17618

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fo 6439  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-map 8617  df-ixp 8686  df-func 17573  df-full 17620

This theorem is referenced by:  ffthiso  17645