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Theorem eldmcoa 17067
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 4874 . 2 (𝐺dom · 𝐹 ↔ ⟨𝐺, 𝐹⟩ ∈ dom · )
2 otex 5154 . . . . . 6 ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩(comp‘𝐶)(coda𝑔))(2nd𝑓))⟩ ∈ V
32rgen2w 3134 . . . . 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 2825 . . . . . . 7 (comp‘𝐶) = (comp‘𝐶)
74, 5, 6coafval 17066 . . . . . 6 · = (𝑔𝐴, 𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩(comp‘𝐶)(coda𝑔))(2nd𝑓))⟩)
87fmpt2x 7499 . . . . 5 (∀𝑔𝐴𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)}⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩(comp‘𝐶)(coda𝑔))(2nd𝑓))⟩ ∈ V ↔ · : 𝑔𝐴 ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)})⟶V)
93, 8mpbi 222 . . . 4 · : 𝑔𝐴 ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)})⟶V
109fdmi 6288 . . 3 dom · = 𝑔𝐴 ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)})
1110eleq2i 2898 . 2 (⟨𝐺, 𝐹⟩ ∈ dom · ↔ ⟨𝐺, 𝐹⟩ ∈ 𝑔𝐴 ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)}))
12 fveq2 6433 . . . . . 6 (𝑔 = 𝐺 → (doma𝑔) = (doma𝐺))
1312eqeq2d 2835 . . . . 5 (𝑔 = 𝐺 → ((coda) = (doma𝑔) ↔ (coda) = (doma𝐺)))
1413rabbidv 3402 . . . 4 (𝑔 = 𝐺 → {𝐴 ∣ (coda) = (doma𝑔)} = {𝐴 ∣ (coda) = (doma𝐺)})
1514opeliunxp2 5493 . . 3 (⟨𝐺, 𝐹⟩ ∈ 𝑔𝐴 ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)}) ↔ (𝐺𝐴𝐹 ∈ {𝐴 ∣ (coda) = (doma𝐺)}))
16 fveqeq2 6442 . . . . 5 ( = 𝐹 → ((coda) = (doma𝐺) ↔ (coda𝐹) = (doma𝐺)))
1716elrab 3585 . . . 4 (𝐹 ∈ {𝐴 ∣ (coda) = (doma𝐺)} ↔ (𝐹𝐴 ∧ (coda𝐹) = (doma𝐺)))
1817anbi2i 618 . . 3 ((𝐺𝐴𝐹 ∈ {𝐴 ∣ (coda) = (doma𝐺)}) ↔ (𝐺𝐴 ∧ (𝐹𝐴 ∧ (coda𝐹) = (doma𝐺))))
19 an12 637 . . . 4 ((𝐺𝐴 ∧ (𝐹𝐴 ∧ (coda𝐹) = (doma𝐺))) ↔ (𝐹𝐴 ∧ (𝐺𝐴 ∧ (coda𝐹) = (doma𝐺))))
20 3anass 1122 . . . 4 ((𝐹𝐴𝐺𝐴 ∧ (coda𝐹) = (doma𝐺)) ↔ (𝐹𝐴 ∧ (𝐺𝐴 ∧ (coda𝐹) = (doma𝐺))))
2119, 20bitr4i 270 . . 3 ((𝐺𝐴 ∧ (𝐹𝐴 ∧ (coda𝐹) = (doma𝐺))) ↔ (𝐹𝐴𝐺𝐴 ∧ (coda𝐹) = (doma𝐺)))
2215, 18, 213bitri 289 . 2 (⟨𝐺, 𝐹⟩ ∈ 𝑔𝐴 ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)}) ↔ (𝐹𝐴𝐺𝐴 ∧ (coda𝐹) = (doma𝐺)))
231, 11, 223bitri 289 1 (𝐺dom · 𝐹 ↔ (𝐹𝐴𝐺𝐴 ∧ (coda𝐹) = (doma𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 386  w3a 1113   = wceq 1658  wcel 2166  wral 3117  {crab 3121  Vcvv 3414  {csn 4397  cop 4403  cotp 4405   ciun 4740   class class class wbr 4873   × cxp 5340  dom cdm 5342  wf 6119  cfv 6123  (class class class)co 6905  2nd c2nd 7427  compcco 16317  domacdoma 17022  codaccoda 17023  Arrowcarw 17024  compaccoa 17056
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-ot 4406  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-1st 7428  df-2nd 7429  df-arw 17029  df-coa 17058
This theorem is referenced by:  homdmcoa  17069  coapm  17073
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