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Theorem eldmcoa 16538
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 4584 . 2 (𝐺dom · 𝐹 ↔ ⟨𝐺, 𝐹⟩ ∈ dom · )
2 otex 4860 . . . . . 6 ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩(comp‘𝐶)(coda𝑔))(2nd𝑓))⟩ ∈ V
32rgen2w 2909 . . . . 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 2610 . . . . . . 7 (comp‘𝐶) = (comp‘𝐶)
74, 5, 6coafval 16537 . . . . . 6 · = (𝑔𝐴, 𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩(comp‘𝐶)(coda𝑔))(2nd𝑓))⟩)
87fmpt2x 7125 . . . . 5 (∀𝑔𝐴𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)}⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩(comp‘𝐶)(coda𝑔))(2nd𝑓))⟩ ∈ V ↔ · : 𝑔𝐴 ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)})⟶V)
93, 8mpbi 219 . . . 4 · : 𝑔𝐴 ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)})⟶V
109fdmi 5965 . . 3 dom · = 𝑔𝐴 ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)})
1110eleq2i 2680 . 2 (⟨𝐺, 𝐹⟩ ∈ dom · ↔ ⟨𝐺, 𝐹⟩ ∈ 𝑔𝐴 ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)}))
12 fveq2 6103 . . . . . 6 (𝑔 = 𝐺 → (doma𝑔) = (doma𝐺))
1312eqeq2d 2620 . . . . 5 (𝑔 = 𝐺 → ((coda) = (doma𝑔) ↔ (coda) = (doma𝐺)))
1413rabbidv 3164 . . . 4 (𝑔 = 𝐺 → {𝐴 ∣ (coda) = (doma𝑔)} = {𝐴 ∣ (coda) = (doma𝐺)})
1514opeliunxp2 5182 . . 3 (⟨𝐺, 𝐹⟩ ∈ 𝑔𝐴 ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)}) ↔ (𝐺𝐴𝐹 ∈ {𝐴 ∣ (coda) = (doma𝐺)}))
16 fveq2 6103 . . . . . 6 ( = 𝐹 → (coda) = (coda𝐹))
1716eqeq1d 2612 . . . . 5 ( = 𝐹 → ((coda) = (doma𝐺) ↔ (coda𝐹) = (doma𝐺)))
1817elrab 3331 . . . 4 (𝐹 ∈ {𝐴 ∣ (coda) = (doma𝐺)} ↔ (𝐹𝐴 ∧ (coda𝐹) = (doma𝐺)))
1918anbi2i 726 . . 3 ((𝐺𝐴𝐹 ∈ {𝐴 ∣ (coda) = (doma𝐺)}) ↔ (𝐺𝐴 ∧ (𝐹𝐴 ∧ (coda𝐹) = (doma𝐺))))
20 an12 834 . . . 4 ((𝐺𝐴 ∧ (𝐹𝐴 ∧ (coda𝐹) = (doma𝐺))) ↔ (𝐹𝐴 ∧ (𝐺𝐴 ∧ (coda𝐹) = (doma𝐺))))
21 3anass 1035 . . . 4 ((𝐹𝐴𝐺𝐴 ∧ (coda𝐹) = (doma𝐺)) ↔ (𝐹𝐴 ∧ (𝐺𝐴 ∧ (coda𝐹) = (doma𝐺))))
2220, 21bitr4i 266 . . 3 ((𝐺𝐴 ∧ (𝐹𝐴 ∧ (coda𝐹) = (doma𝐺))) ↔ (𝐹𝐴𝐺𝐴 ∧ (coda𝐹) = (doma𝐺)))
2315, 19, 223bitri 285 . 2 (⟨𝐺, 𝐹⟩ ∈ 𝑔𝐴 ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)}) ↔ (𝐹𝐴𝐺𝐴 ∧ (coda𝐹) = (doma𝐺)))
241, 11, 233bitri 285 1 (𝐺dom · 𝐹 ↔ (𝐹𝐴𝐺𝐴 ∧ (coda𝐹) = (doma𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wral 2896  {crab 2900  Vcvv 3173  {csn 4125  cop 4131  cotp 4133   ciun 4455   class class class wbr 4583   × cxp 5036  dom cdm 5038  wf 5800  cfv 5804  (class class class)co 6549  2nd c2nd 7058  compcco 15780  domacdoma 16493  codaccoda 16494  Arrowcarw 16495  compaccoa 16527
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-ot 4134  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-arw 16500  df-coa 16529
This theorem is referenced by:  homdmcoa  16540  coapm  16544
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