Theorem fulli 17976

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 17975	. 2 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–onto→((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
8		fulli.r	. 2 ⊢ (𝜑 → 𝑅 ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
9		foelrn 7134	. 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 2108  ∃wrex 3070   class class class wbr 5151  –onto→wfo 6567  ‘cfv 6569  (class class class)co 7438  Basecbs 17254  Hom chom 17318   Full cful 17965

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5288  ax-sep 5305  ax-nul 5315  ax-pow 5374  ax-pr 5441  ax-un 7761

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3483  df-sbc 3795  df-csb 3912  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-nul 4343  df-if 4535  df-pw 4610  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-iun 5001  df-br 5152  df-opab 5214  df-mpt 5235  df-id 5587  df-xp 5699  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-rn 5704  df-res 5705  df-ima 5706  df-iota 6522  df-fun 6571  df-fn 6572  df-f 6573  df-fo 6575  df-fv 6577  df-ov 7441  df-oprab 7442  df-mpo 7443  df-1st 8022  df-2nd 8023  df-map 8876  df-ixp 8946  df-func 17918  df-full 17967

This theorem is referenced by:  ffthiso  17992