Theorem fthi 17872

Description: The morphism map of a faithful functor is an injection. (Contributed by Mario Carneiro, 27-Jan-2017.)

Hypotheses

Ref	Expression
isfth.b	⊢ 𝐵 = (Base‘𝐶)
isfth.h	⊢ 𝐻 = (Hom ‘𝐶)
isfth.j	⊢ 𝐽 = (Hom ‘𝐷)
fthf1.f	⊢ (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺)
fthf1.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
fthf1.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
fthi.r	⊢ (𝜑 → 𝑅 ∈ (𝑋𝐻𝑌))
fthi.s	⊢ (𝜑 → 𝑆 ∈ (𝑋𝐻𝑌))

Assertion

Ref	Expression
fthi	⊢ (𝜑 → (((𝑋𝐺𝑌)‘𝑅) = ((𝑋𝐺𝑌)‘𝑆) ↔ 𝑅 = 𝑆))

Proof of Theorem fthi

Step	Hyp	Ref	Expression
1		isfth.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
2		isfth.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐶)
3		isfth.j	. . 3 ⊢ 𝐽 = (Hom ‘𝐷)
4		fthf1.f	. . 3 ⊢ (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺)
5		fthf1.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
6		fthf1.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
7	1, 2, 3, 4, 5, 6	fthf1 17871	. 2 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1→((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
8		fthi.r	. 2 ⊢ (𝜑 → 𝑅 ∈ (𝑋𝐻𝑌))
9		fthi.s	. 2 ⊢ (𝜑 → 𝑆 ∈ (𝑋𝐻𝑌))
10		f1fveq 7254	. 2 ⊢ (((𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1→((𝐹‘𝑋)𝐽(𝐹‘𝑌)) ∧ (𝑅 ∈ (𝑋𝐻𝑌) ∧ 𝑆 ∈ (𝑋𝐻𝑌))) → (((𝑋𝐺𝑌)‘𝑅) = ((𝑋𝐺𝑌)‘𝑆) ↔ 𝑅 = 𝑆))
11	7, 8, 9, 10	syl12anc 834	1 ⊢ (𝜑 → (((𝑋𝐺𝑌)‘𝑅) = ((𝑋𝐺𝑌)‘𝑆) ↔ 𝑅 = 𝑆))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1533   ∈ wcel 2098   class class class wbr 5139  –1-1→wf1 6531  ‘cfv 6534  (class class class)co 7402  Basecbs 17145  Hom chom 17209   Faith cfth 17857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-1st 7969  df-2nd 7970  df-map 8819  df-ixp 8889  df-func 17809  df-fth 17859

This theorem is referenced by:  fthsect  17879  fthmon  17881