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Theorem dmfco 5455
Description: Domains of a function composition. (Contributed by NM, 27-Jan-1997.)
Assertion
Ref Expression
dmfco ((Fun 𝐺𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹𝐺) ↔ (𝐺𝐴) ∈ dom 𝐹))

Proof of Theorem dmfco
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funfvex 5404 . . . . 5 ((Fun 𝐺𝐴 ∈ dom 𝐺) → (𝐺𝐴) ∈ V)
2 opeq1 3673 . . . . . . 7 (𝑥 = (𝐺𝐴) → ⟨𝑥, 𝑦⟩ = ⟨(𝐺𝐴), 𝑦⟩)
32eleq1d 2184 . . . . . 6 (𝑥 = (𝐺𝐴) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨(𝐺𝐴), 𝑦⟩ ∈ 𝐹))
43ceqsexgv 2786 . . . . 5 ((𝐺𝐴) ∈ V → (∃𝑥(𝑥 = (𝐺𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ↔ ⟨(𝐺𝐴), 𝑦⟩ ∈ 𝐹))
51, 4syl 14 . . . 4 ((Fun 𝐺𝐴 ∈ dom 𝐺) → (∃𝑥(𝑥 = (𝐺𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ↔ ⟨(𝐺𝐴), 𝑦⟩ ∈ 𝐹))
6 eqcom 2117 . . . . . . 7 (𝑥 = (𝐺𝐴) ↔ (𝐺𝐴) = 𝑥)
7 funopfvb 5431 . . . . . . 7 ((Fun 𝐺𝐴 ∈ dom 𝐺) → ((𝐺𝐴) = 𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐺))
86, 7syl5bb 191 . . . . . 6 ((Fun 𝐺𝐴 ∈ dom 𝐺) → (𝑥 = (𝐺𝐴) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐺))
98anbi1d 458 . . . . 5 ((Fun 𝐺𝐴 ∈ dom 𝐺) → ((𝑥 = (𝐺𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ↔ (⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
109exbidv 1779 . . . 4 ((Fun 𝐺𝐴 ∈ dom 𝐺) → (∃𝑥(𝑥 = (𝐺𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
115, 10bitr3d 189 . . 3 ((Fun 𝐺𝐴 ∈ dom 𝐺) → (⟨(𝐺𝐴), 𝑦⟩ ∈ 𝐹 ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
1211exbidv 1779 . 2 ((Fun 𝐺𝐴 ∈ dom 𝐺) → (∃𝑦⟨(𝐺𝐴), 𝑦⟩ ∈ 𝐹 ↔ ∃𝑦𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
13 eldm2g 4703 . . 3 ((𝐺𝐴) ∈ V → ((𝐺𝐴) ∈ dom 𝐹 ↔ ∃𝑦⟨(𝐺𝐴), 𝑦⟩ ∈ 𝐹))
141, 13syl 14 . 2 ((Fun 𝐺𝐴 ∈ dom 𝐺) → ((𝐺𝐴) ∈ dom 𝐹 ↔ ∃𝑦⟨(𝐺𝐴), 𝑦⟩ ∈ 𝐹))
15 eldm2g 4703 . . . 4 (𝐴 ∈ dom 𝐺 → (𝐴 ∈ dom (𝐹𝐺) ↔ ∃𝑦𝐴, 𝑦⟩ ∈ (𝐹𝐺)))
16 vex 2661 . . . . . 6 𝑦 ∈ V
17 opelco2g 4675 . . . . . 6 ((𝐴 ∈ dom 𝐺𝑦 ∈ V) → (⟨𝐴, 𝑦⟩ ∈ (𝐹𝐺) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
1816, 17mpan2 419 . . . . 5 (𝐴 ∈ dom 𝐺 → (⟨𝐴, 𝑦⟩ ∈ (𝐹𝐺) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
1918exbidv 1779 . . . 4 (𝐴 ∈ dom 𝐺 → (∃𝑦𝐴, 𝑦⟩ ∈ (𝐹𝐺) ↔ ∃𝑦𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
2015, 19bitrd 187 . . 3 (𝐴 ∈ dom 𝐺 → (𝐴 ∈ dom (𝐹𝐺) ↔ ∃𝑦𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
2120adantl 273 . 2 ((Fun 𝐺𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹𝐺) ↔ ∃𝑦𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
2212, 14, 213bitr4rd 220 1 ((Fun 𝐺𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹𝐺) ↔ (𝐺𝐴) ∈ dom 𝐹))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1314  wex 1451  wcel 1463  Vcvv 2658  cop 3498  dom cdm 4507  ccom 4511  Fun wfun 5085  cfv 5091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-iota 5056  df-fun 5093  df-fn 5094  df-fv 5099
This theorem is referenced by:  ctssdccl  6962
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