Theorem fthi 17955

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 17954	. 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 7248	. 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 847	1 ⊢ (𝜑 → (((𝑋𝐺𝑌)‘𝑅) = ((𝑋𝐺𝑌)‘𝑆) ↔ 𝑅 = 𝑆))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1562   ∈ wcel 2144   class class class wbr 5102  –1-1→wf1 6520  ‘cfv 6523  (class class class)co 7398  Basecbs 17247  Hom chom 17299   Faith cfth 17940

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fv 6531  df-ov 7401  df-oprab 7402  df-mpo 7403  df-1st 7972  df-2nd 7973  df-map 8812  df-ixp 8882  df-func 17893  df-fth 17942

This theorem is referenced by:  fthsect  17962  fthmon  17964  fthcomf  49783