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Theorem eldmcoa 17958
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 Β· = (compaβ€˜πΆ)
coafval.a 𝐴 = (Arrowβ€˜πΆ)
Assertion
Ref Expression
eldmcoa (𝐺dom Β· 𝐹 ↔ (𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴 ∧ (codaβ€˜πΉ) = (domaβ€˜πΊ)))

Proof of Theorem eldmcoa
Dummy variables 𝑓 𝑔 β„Ž are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-br 5111 . 2 (𝐺dom Β· 𝐹 ↔ ⟨𝐺, 𝐹⟩ ∈ dom Β· )
2 otex 5427 . . . . . 6 ⟨(domaβ€˜π‘“), (codaβ€˜π‘”), ((2nd β€˜π‘”)(⟨(domaβ€˜π‘“), (domaβ€˜π‘”)⟩(compβ€˜πΆ)(codaβ€˜π‘”))(2nd β€˜π‘“))⟩ ∈ V
32rgen2w 3070 . . . . 5 βˆ€π‘” ∈ 𝐴 βˆ€π‘“ ∈ {β„Ž ∈ 𝐴 ∣ (codaβ€˜β„Ž) = (domaβ€˜π‘”)}⟨(domaβ€˜π‘“), (codaβ€˜π‘”), ((2nd β€˜π‘”)(⟨(domaβ€˜π‘“), (domaβ€˜π‘”)⟩(compβ€˜πΆ)(codaβ€˜π‘”))(2nd β€˜π‘“))⟩ ∈ V
4 coafval.o . . . . . . 7 Β· = (compaβ€˜πΆ)
5 coafval.a . . . . . . 7 𝐴 = (Arrowβ€˜πΆ)
6 eqid 2737 . . . . . . 7 (compβ€˜πΆ) = (compβ€˜πΆ)
74, 5, 6coafval 17957 . . . . . 6 Β· = (𝑔 ∈ 𝐴, 𝑓 ∈ {β„Ž ∈ 𝐴 ∣ (codaβ€˜β„Ž) = (domaβ€˜π‘”)} ↦ ⟨(domaβ€˜π‘“), (codaβ€˜π‘”), ((2nd β€˜π‘”)(⟨(domaβ€˜π‘“), (domaβ€˜π‘”)⟩(compβ€˜πΆ)(codaβ€˜π‘”))(2nd β€˜π‘“))⟩)
87fmpox 8004 . . . . 5 (βˆ€π‘” ∈ 𝐴 βˆ€π‘“ ∈ {β„Ž ∈ 𝐴 ∣ (codaβ€˜β„Ž) = (domaβ€˜π‘”)}⟨(domaβ€˜π‘“), (codaβ€˜π‘”), ((2nd β€˜π‘”)(⟨(domaβ€˜π‘“), (domaβ€˜π‘”)⟩(compβ€˜πΆ)(codaβ€˜π‘”))(2nd β€˜π‘“))⟩ ∈ V ↔ Β· :βˆͺ 𝑔 ∈ 𝐴 ({𝑔} Γ— {β„Ž ∈ 𝐴 ∣ (codaβ€˜β„Ž) = (domaβ€˜π‘”)})⟢V)
93, 8mpbi 229 . . . 4 Β· :βˆͺ 𝑔 ∈ 𝐴 ({𝑔} Γ— {β„Ž ∈ 𝐴 ∣ (codaβ€˜β„Ž) = (domaβ€˜π‘”)})⟢V
109fdmi 6685 . . 3 dom Β· = βˆͺ 𝑔 ∈ 𝐴 ({𝑔} Γ— {β„Ž ∈ 𝐴 ∣ (codaβ€˜β„Ž) = (domaβ€˜π‘”)})
1110eleq2i 2830 . 2 (⟨𝐺, 𝐹⟩ ∈ dom Β· ↔ ⟨𝐺, 𝐹⟩ ∈ βˆͺ 𝑔 ∈ 𝐴 ({𝑔} Γ— {β„Ž ∈ 𝐴 ∣ (codaβ€˜β„Ž) = (domaβ€˜π‘”)}))
12 fveq2 6847 . . . . . 6 (𝑔 = 𝐺 β†’ (domaβ€˜π‘”) = (domaβ€˜πΊ))
1312eqeq2d 2748 . . . . 5 (𝑔 = 𝐺 β†’ ((codaβ€˜β„Ž) = (domaβ€˜π‘”) ↔ (codaβ€˜β„Ž) = (domaβ€˜πΊ)))
1413rabbidv 3418 . . . 4 (𝑔 = 𝐺 β†’ {β„Ž ∈ 𝐴 ∣ (codaβ€˜β„Ž) = (domaβ€˜π‘”)} = {β„Ž ∈ 𝐴 ∣ (codaβ€˜β„Ž) = (domaβ€˜πΊ)})
1514opeliunxp2 5799 . . 3 (⟨𝐺, 𝐹⟩ ∈ βˆͺ 𝑔 ∈ 𝐴 ({𝑔} Γ— {β„Ž ∈ 𝐴 ∣ (codaβ€˜β„Ž) = (domaβ€˜π‘”)}) ↔ (𝐺 ∈ 𝐴 ∧ 𝐹 ∈ {β„Ž ∈ 𝐴 ∣ (codaβ€˜β„Ž) = (domaβ€˜πΊ)}))
16 fveqeq2 6856 . . . . 5 (β„Ž = 𝐹 β†’ ((codaβ€˜β„Ž) = (domaβ€˜πΊ) ↔ (codaβ€˜πΉ) = (domaβ€˜πΊ)))
1716elrab 3650 . . . 4 (𝐹 ∈ {β„Ž ∈ 𝐴 ∣ (codaβ€˜β„Ž) = (domaβ€˜πΊ)} ↔ (𝐹 ∈ 𝐴 ∧ (codaβ€˜πΉ) = (domaβ€˜πΊ)))
1817anbi2i 624 . . 3 ((𝐺 ∈ 𝐴 ∧ 𝐹 ∈ {β„Ž ∈ 𝐴 ∣ (codaβ€˜β„Ž) = (domaβ€˜πΊ)}) ↔ (𝐺 ∈ 𝐴 ∧ (𝐹 ∈ 𝐴 ∧ (codaβ€˜πΉ) = (domaβ€˜πΊ))))
19 an12 644 . . . 4 ((𝐺 ∈ 𝐴 ∧ (𝐹 ∈ 𝐴 ∧ (codaβ€˜πΉ) = (domaβ€˜πΊ))) ↔ (𝐹 ∈ 𝐴 ∧ (𝐺 ∈ 𝐴 ∧ (codaβ€˜πΉ) = (domaβ€˜πΊ))))
20 3anass 1096 . . . 4 ((𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴 ∧ (codaβ€˜πΉ) = (domaβ€˜πΊ)) ↔ (𝐹 ∈ 𝐴 ∧ (𝐺 ∈ 𝐴 ∧ (codaβ€˜πΉ) = (domaβ€˜πΊ))))
2119, 20bitr4i 278 . . 3 ((𝐺 ∈ 𝐴 ∧ (𝐹 ∈ 𝐴 ∧ (codaβ€˜πΉ) = (domaβ€˜πΊ))) ↔ (𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴 ∧ (codaβ€˜πΉ) = (domaβ€˜πΊ)))
2215, 18, 213bitri 297 . 2 (⟨𝐺, 𝐹⟩ ∈ βˆͺ 𝑔 ∈ 𝐴 ({𝑔} Γ— {β„Ž ∈ 𝐴 ∣ (codaβ€˜β„Ž) = (domaβ€˜π‘”)}) ↔ (𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴 ∧ (codaβ€˜πΉ) = (domaβ€˜πΊ)))
231, 11, 223bitri 297 1 (𝐺dom Β· 𝐹 ↔ (𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴 ∧ (codaβ€˜πΉ) = (domaβ€˜πΊ)))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  {crab 3410  Vcvv 3448  {csn 4591  βŸ¨cop 4597  βŸ¨cotp 4599  βˆͺ ciun 4959   class class class wbr 5110   Γ— cxp 5636  dom cdm 5638  βŸΆwf 6497  β€˜cfv 6501  (class class class)co 7362  2nd c2nd 7925  compcco 17152  domacdoma 17913  codaccoda 17914  Arrowcarw 17915  compaccoa 17947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-ot 4600  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-1st 7926  df-2nd 7927  df-arw 17920  df-coa 17949
This theorem is referenced by:  homdmcoa  17960  coapm  17964
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