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Theorem eldmcoa 16921
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 4785 . 2 (𝐺dom · 𝐹 ↔ ⟨𝐺, 𝐹⟩ ∈ dom · )
2 otex 5061 . . . . . 6 ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩(comp‘𝐶)(coda𝑔))(2nd𝑓))⟩ ∈ V
32rgen2w 3073 . . . . 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 2770 . . . . . . 7 (comp‘𝐶) = (comp‘𝐶)
74, 5, 6coafval 16920 . . . . . 6 · = (𝑔𝐴, 𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩(comp‘𝐶)(coda𝑔))(2nd𝑓))⟩)
87fmpt2x 7385 . . . . 5 (∀𝑔𝐴𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)}⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩(comp‘𝐶)(coda𝑔))(2nd𝑓))⟩ ∈ V ↔ · : 𝑔𝐴 ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)})⟶V)
93, 8mpbi 220 . . . 4 · : 𝑔𝐴 ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)})⟶V
109fdmi 6192 . . 3 dom · = 𝑔𝐴 ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)})
1110eleq2i 2841 . 2 (⟨𝐺, 𝐹⟩ ∈ dom · ↔ ⟨𝐺, 𝐹⟩ ∈ 𝑔𝐴 ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)}))
12 fveq2 6332 . . . . . 6 (𝑔 = 𝐺 → (doma𝑔) = (doma𝐺))
1312eqeq2d 2780 . . . . 5 (𝑔 = 𝐺 → ((coda) = (doma𝑔) ↔ (coda) = (doma𝐺)))
1413rabbidv 3338 . . . 4 (𝑔 = 𝐺 → {𝐴 ∣ (coda) = (doma𝑔)} = {𝐴 ∣ (coda) = (doma𝐺)})
1514opeliunxp2 5399 . . 3 (⟨𝐺, 𝐹⟩ ∈ 𝑔𝐴 ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)}) ↔ (𝐺𝐴𝐹 ∈ {𝐴 ∣ (coda) = (doma𝐺)}))
16 fveq2 6332 . . . . . 6 ( = 𝐹 → (coda) = (coda𝐹))
1716eqeq1d 2772 . . . . 5 ( = 𝐹 → ((coda) = (doma𝐺) ↔ (coda𝐹) = (doma𝐺)))
1817elrab 3513 . . . 4 (𝐹 ∈ {𝐴 ∣ (coda) = (doma𝐺)} ↔ (𝐹𝐴 ∧ (coda𝐹) = (doma𝐺)))
1918anbi2i 601 . . 3 ((𝐺𝐴𝐹 ∈ {𝐴 ∣ (coda) = (doma𝐺)}) ↔ (𝐺𝐴 ∧ (𝐹𝐴 ∧ (coda𝐹) = (doma𝐺))))
20 an12 616 . . . 4 ((𝐺𝐴 ∧ (𝐹𝐴 ∧ (coda𝐹) = (doma𝐺))) ↔ (𝐹𝐴 ∧ (𝐺𝐴 ∧ (coda𝐹) = (doma𝐺))))
21 3anass 1079 . . . 4 ((𝐹𝐴𝐺𝐴 ∧ (coda𝐹) = (doma𝐺)) ↔ (𝐹𝐴 ∧ (𝐺𝐴 ∧ (coda𝐹) = (doma𝐺))))
2220, 21bitr4i 267 . . 3 ((𝐺𝐴 ∧ (𝐹𝐴 ∧ (coda𝐹) = (doma𝐺))) ↔ (𝐹𝐴𝐺𝐴 ∧ (coda𝐹) = (doma𝐺)))
2315, 19, 223bitri 286 . 2 (⟨𝐺, 𝐹⟩ ∈ 𝑔𝐴 ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)}) ↔ (𝐹𝐴𝐺𝐴 ∧ (coda𝐹) = (doma𝐺)))
241, 11, 233bitri 286 1 (𝐺dom · 𝐹 ↔ (𝐹𝐴𝐺𝐴 ∧ (coda𝐹) = (doma𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382  w3a 1070   = wceq 1630  wcel 2144  wral 3060  {crab 3064  Vcvv 3349  {csn 4314  cop 4320  cotp 4322   ciun 4652   class class class wbr 4784   × cxp 5247  dom cdm 5249  wf 6027  cfv 6031  (class class class)co 6792  2nd c2nd 7313  compcco 16160  domacdoma 16876  codaccoda 16877  Arrowcarw 16878  compaccoa 16910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-ot 4323  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-1st 7314  df-2nd 7315  df-arw 16883  df-coa 16912
This theorem is referenced by:  homdmcoa  16923  coapm  16927
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