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

Theorem eldmcoa 17313
Description: A pair 𝐺, 𝐹 is in the domain of the arrow composition, if the domain of 𝐺 equals the codomain of 𝐹. (In this case we say 𝐺 and 𝐹 are composable.) (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
coafval.o · = (compa𝐶)
coafval.a 𝐴 = (Arrow‘𝐶)
Assertion
Ref Expression
eldmcoa (𝐺dom · 𝐹 ↔ (𝐹𝐴𝐺𝐴 ∧ (coda𝐹) = (doma𝐺)))

Proof of Theorem eldmcoa
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-br 5058 . 2 (𝐺dom · 𝐹 ↔ ⟨𝐺, 𝐹⟩ ∈ dom · )
2 otex 5348 . . . . . 6 ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩(comp‘𝐶)(coda𝑔))(2nd𝑓))⟩ ∈ V
32rgen2w 3148 . . . . 5 𝑔𝐴𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)}⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩(comp‘𝐶)(coda𝑔))(2nd𝑓))⟩ ∈ V
4 coafval.o . . . . . . 7 · = (compa𝐶)
5 coafval.a . . . . . . 7 𝐴 = (Arrow‘𝐶)
6 eqid 2818 . . . . . . 7 (comp‘𝐶) = (comp‘𝐶)
74, 5, 6coafval 17312 . . . . . 6 · = (𝑔𝐴, 𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩(comp‘𝐶)(coda𝑔))(2nd𝑓))⟩)
87fmpox 7754 . . . . 5 (∀𝑔𝐴𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)}⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩(comp‘𝐶)(coda𝑔))(2nd𝑓))⟩ ∈ V ↔ · : 𝑔𝐴 ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)})⟶V)
93, 8mpbi 231 . . . 4 · : 𝑔𝐴 ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)})⟶V
109fdmi 6517 . . 3 dom · = 𝑔𝐴 ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)})
1110eleq2i 2901 . 2 (⟨𝐺, 𝐹⟩ ∈ dom · ↔ ⟨𝐺, 𝐹⟩ ∈ 𝑔𝐴 ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)}))
12 fveq2 6663 . . . . . 6 (𝑔 = 𝐺 → (doma𝑔) = (doma𝐺))
1312eqeq2d 2829 . . . . 5 (𝑔 = 𝐺 → ((coda) = (doma𝑔) ↔ (coda) = (doma𝐺)))
1413rabbidv 3478 . . . 4 (𝑔 = 𝐺 → {𝐴 ∣ (coda) = (doma𝑔)} = {𝐴 ∣ (coda) = (doma𝐺)})
1514opeliunxp2 5702 . . 3 (⟨𝐺, 𝐹⟩ ∈ 𝑔𝐴 ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)}) ↔ (𝐺𝐴𝐹 ∈ {𝐴 ∣ (coda) = (doma𝐺)}))
16 fveqeq2 6672 . . . . 5 ( = 𝐹 → ((coda) = (doma𝐺) ↔ (coda𝐹) = (doma𝐺)))
1716elrab 3677 . . . 4 (𝐹 ∈ {𝐴 ∣ (coda) = (doma𝐺)} ↔ (𝐹𝐴 ∧ (coda𝐹) = (doma𝐺)))
1817anbi2i 622 . . 3 ((𝐺𝐴𝐹 ∈ {𝐴 ∣ (coda) = (doma𝐺)}) ↔ (𝐺𝐴 ∧ (𝐹𝐴 ∧ (coda𝐹) = (doma𝐺))))
19 an12 641 . . . 4 ((𝐺𝐴 ∧ (𝐹𝐴 ∧ (coda𝐹) = (doma𝐺))) ↔ (𝐹𝐴 ∧ (𝐺𝐴 ∧ (coda𝐹) = (doma𝐺))))
20 3anass 1087 . . . 4 ((𝐹𝐴𝐺𝐴 ∧ (coda𝐹) = (doma𝐺)) ↔ (𝐹𝐴 ∧ (𝐺𝐴 ∧ (coda𝐹) = (doma𝐺))))
2119, 20bitr4i 279 . . 3 ((𝐺𝐴 ∧ (𝐹𝐴 ∧ (coda𝐹) = (doma𝐺))) ↔ (𝐹𝐴𝐺𝐴 ∧ (coda𝐹) = (doma𝐺)))
2215, 18, 213bitri 298 . 2 (⟨𝐺, 𝐹⟩ ∈ 𝑔𝐴 ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)}) ↔ (𝐹𝐴𝐺𝐴 ∧ (coda𝐹) = (doma𝐺)))
231, 11, 223bitri 298 1 (𝐺dom · 𝐹 ↔ (𝐹𝐴𝐺𝐴 ∧ (coda𝐹) = (doma𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  {crab 3139  Vcvv 3492  {csn 4557  cop 4563  cotp 4565   ciun 4910   class class class wbr 5057   × cxp 5546  dom cdm 5548  wf 6344  cfv 6348  (class class class)co 7145  2nd c2nd 7677  compcco 16565  domacdoma 17268  codaccoda 17269  Arrowcarw 17270  compaccoa 17302
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-ot 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-arw 17275  df-coa 17304
This theorem is referenced by:  homdmcoa  17315  coapm  17319
  Copyright terms: Public domain W3C validator