Theorem homfeqd 17869

Description: If two structures have the same Hom slot, they have the same Hom-sets. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
homfeqd.1	⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))
homfeqd.2	⊢ (𝜑 → (Hom ‘𝐶) = (Hom ‘𝐷))

Assertion

Ref	Expression
homfeqd	⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))

Proof of Theorem homfeqd

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		homfeqd.2	. . . . 5 ⊢ (𝜑 → (Hom ‘𝐶) = (Hom ‘𝐷))
2	1	oveqd 7437	. . . 4 ⊢ (𝜑 → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
3	2	ralrimivw 3158	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ (Base‘𝐶)(𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
4	3	ralrimivw 3158	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
5		eqid 2760	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
6		eqid 2760	. . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
7		eqidd 2761	. . 3 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐶))
8		homfeqd.1	. . 3 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))
9	5, 6, 7, 8	homfeq 17868	. 2 ⊢ (𝜑 → ((Homf ‘𝐶) = (Homf ‘𝐷) ↔ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦)))
10	4, 9	mpbird 260	1 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))

This proof depends on syntax axioms:   → wi 4   = wceq 1569  ∀wral 3076  ‘cfv 6538  (class class class)co 7420  Basecbs 17387  Hom chom 17439  Hom_f chomf 17840

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-homf 17844

This theorem is used by: (None)