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Theorem dmfco 5584
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 5532 . . . . 5 ((Fun 𝐺𝐴 ∈ dom 𝐺) → (𝐺𝐴) ∈ V)
2 opeq1 3778 . . . . . . 7 (𝑥 = (𝐺𝐴) → ⟨𝑥, 𝑦⟩ = ⟨(𝐺𝐴), 𝑦⟩)
32eleq1d 2246 . . . . . 6 (𝑥 = (𝐺𝐴) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨(𝐺𝐴), 𝑦⟩ ∈ 𝐹))
43ceqsexgv 2866 . . . . 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 2179 . . . . . . 7 (𝑥 = (𝐺𝐴) ↔ (𝐺𝐴) = 𝑥)
7 funopfvb 5559 . . . . . . 7 ((Fun 𝐺𝐴 ∈ dom 𝐺) → ((𝐺𝐴) = 𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐺))
86, 7bitrid 192 . . . . . 6 ((Fun 𝐺𝐴 ∈ dom 𝐺) → (𝑥 = (𝐺𝐴) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐺))
98anbi1d 465 . . . . 5 ((Fun 𝐺𝐴 ∈ dom 𝐺) → ((𝑥 = (𝐺𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ↔ (⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
109exbidv 1825 . . . 4 ((Fun 𝐺𝐴 ∈ dom 𝐺) → (∃𝑥(𝑥 = (𝐺𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
115, 10bitr3d 190 . . 3 ((Fun 𝐺𝐴 ∈ dom 𝐺) → (⟨(𝐺𝐴), 𝑦⟩ ∈ 𝐹 ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
1211exbidv 1825 . 2 ((Fun 𝐺𝐴 ∈ dom 𝐺) → (∃𝑦⟨(𝐺𝐴), 𝑦⟩ ∈ 𝐹 ↔ ∃𝑦𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
13 eldm2g 4823 . . 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 4823 . . . 4 (𝐴 ∈ dom 𝐺 → (𝐴 ∈ dom (𝐹𝐺) ↔ ∃𝑦𝐴, 𝑦⟩ ∈ (𝐹𝐺)))
16 vex 2740 . . . . . 6 𝑦 ∈ V
17 opelco2g 4795 . . . . . 6 ((𝐴 ∈ dom 𝐺𝑦 ∈ V) → (⟨𝐴, 𝑦⟩ ∈ (𝐹𝐺) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
1816, 17mpan2 425 . . . . 5 (𝐴 ∈ dom 𝐺 → (⟨𝐴, 𝑦⟩ ∈ (𝐹𝐺) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
1918exbidv 1825 . . . 4 (𝐴 ∈ dom 𝐺 → (∃𝑦𝐴, 𝑦⟩ ∈ (𝐹𝐺) ↔ ∃𝑦𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
2015, 19bitrd 188 . . 3 (𝐴 ∈ dom 𝐺 → (𝐴 ∈ dom (𝐹𝐺) ↔ ∃𝑦𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
2120adantl 277 . 2 ((Fun 𝐺𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹𝐺) ↔ ∃𝑦𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
2212, 14, 213bitr4rd 221 1 ((Fun 𝐺𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹𝐺) ↔ (𝐺𝐴) ∈ dom 𝐹))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wex 1492  wcel 2148  Vcvv 2737  cop 3595  dom cdm 4626  ccom 4630  Fun wfun 5210  cfv 5216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-iota 5178  df-fun 5218  df-fn 5219  df-fv 5224
This theorem is referenced by:  ctssdccl  7109
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