Theorem fulli 17931

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 17930	. 2 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–onto→((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
8		fulli.r	. 2 ⊢ (𝜑 → 𝑅 ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
9		foelrn 7107	. 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 2107  ∃wrex 3059   class class class wbr 5123  –onto→wfo 6539  ‘cfv 6541  (class class class)co 7413  Basecbs 17229  Hom chom 17284   Full cful 17920

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5259  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-fo 6547  df-fv 6549  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7996  df-2nd 7997  df-map 8850  df-ixp 8920  df-func 17874  df-full 17922

This theorem is referenced by:  ffthiso  17947