Theorem fulli 17960

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 17959	. 2 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–onto→((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
8		fulli.r	. 2 ⊢ (𝜑 → 𝑅 ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
9		foelrn 7092	. 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 595	1 ⊢ (𝜑 → ∃𝑓 ∈ (𝑋𝐻𝑌)𝑅 = ((𝑋𝐺𝑌)‘𝑓))

Syntax hints:   → wi 4   = wceq 1563   ∈ wcel 2145  ∃wrex 3089   class class class wbr 5104  –onto→wfo 6523  ‘cfv 6525  (class class class)co 7400  Basecbs 17257  Hom chom 17309   Full cful 17949

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5231  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-fo 6531  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975  df-map 8814  df-ixp 8884  df-func 17903  df-full 17951

This theorem is referenced by:  ffthiso  17976