Theorem fthi 18095

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 18094	. 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 7266	. 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 850	1 ⊢ (𝜑 → (((𝑋𝐺𝑌)‘𝑅) = ((𝑋𝐺𝑌)‘𝑆) ↔ 𝑅 = 𝑆))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2145   class class class wbr 5103  –1-1→wf1 6535  ‘cfv 6538  (class class class)co 7420  Basecbs 17387  Hom chom 17439   Faith cfth 18080

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  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 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-map 8849  df-ixp 8926  df-func 18033  df-fth 18082

This theorem is used by:  fthsect  18102  fthmon  18104  fthcomf  50264