MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  homfeq Structured version   Visualization version   GIF version

Theorem homfeq 16177
Description: Condition for two categories with the same base to have the same hom-sets. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
homfeq.h 𝐻 = (Hom ‘𝐶)
homfeq.j 𝐽 = (Hom ‘𝐷)
homfeq.1 (𝜑𝐵 = (Base‘𝐶))
homfeq.2 (𝜑𝐵 = (Base‘𝐷))
Assertion
Ref Expression
homfeq (𝜑 → ((Homf𝐶) = (Homf𝐷) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝐻𝑦) = (𝑥𝐽𝑦)))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦   𝑥,𝐻,𝑦   𝜑,𝑥,𝑦   𝑥,𝐽,𝑦

Proof of Theorem homfeq
StepHypRef Expression
1 homfeq.1 . . . . 5 (𝜑𝐵 = (Base‘𝐶))
2 eqidd 2611 . . . . 5 (𝜑 → (𝑥𝐻𝑦) = (𝑥𝐻𝑦))
31, 1, 2mpt2eq123dv 6615 . . . 4 (𝜑 → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥𝐻𝑦)))
4 eqid 2610 . . . . 5 (Homf𝐶) = (Homf𝐶)
5 eqid 2610 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
6 homfeq.h . . . . 5 𝐻 = (Hom ‘𝐶)
74, 5, 6homffval 16173 . . . 4 (Homf𝐶) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥𝐻𝑦))
83, 7syl6reqr 2663 . . 3 (𝜑 → (Homf𝐶) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)))
9 homfeq.2 . . . . 5 (𝜑𝐵 = (Base‘𝐷))
10 eqidd 2611 . . . . 5 (𝜑 → (𝑥𝐽𝑦) = (𝑥𝐽𝑦))
119, 9, 10mpt2eq123dv 6615 . . . 4 (𝜑 → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐽𝑦)) = (𝑥 ∈ (Base‘𝐷), 𝑦 ∈ (Base‘𝐷) ↦ (𝑥𝐽𝑦)))
12 eqid 2610 . . . . 5 (Homf𝐷) = (Homf𝐷)
13 eqid 2610 . . . . 5 (Base‘𝐷) = (Base‘𝐷)
14 homfeq.j . . . . 5 𝐽 = (Hom ‘𝐷)
1512, 13, 14homffval 16173 . . . 4 (Homf𝐷) = (𝑥 ∈ (Base‘𝐷), 𝑦 ∈ (Base‘𝐷) ↦ (𝑥𝐽𝑦))
1611, 15syl6reqr 2663 . . 3 (𝜑 → (Homf𝐷) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐽𝑦)))
178, 16eqeq12d 2625 . 2 (𝜑 → ((Homf𝐶) = (Homf𝐷) ↔ (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐽𝑦))))
18 ovex 6577 . . . 4 (𝑥𝐻𝑦) ∈ V
1918rgen2w 2909 . . 3 𝑥𝐵𝑦𝐵 (𝑥𝐻𝑦) ∈ V
20 mpt22eqb 6667 . . 3 (∀𝑥𝐵𝑦𝐵 (𝑥𝐻𝑦) ∈ V → ((𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐽𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝐻𝑦) = (𝑥𝐽𝑦)))
2119, 20ax-mp 5 . 2 ((𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐽𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝐻𝑦) = (𝑥𝐽𝑦))
2217, 21syl6bb 275 1 (𝜑 → ((Homf𝐶) = (Homf𝐷) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝐻𝑦) = (𝑥𝐽𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195   = wceq 1475  wcel 1977  wral 2896  Vcvv 3173  cfv 5804  (class class class)co 6549  cmpt2 6551  Basecbs 15695  Hom chom 15779  Homf chomf 16150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-homf 16154
This theorem is referenced by:  homfeqd  16178  fullresc  16334  resssetc  16565  resscatc  16578
  Copyright terms: Public domain W3C validator