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Theorem dmfco 6934
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 eldm2g 5853 . . . 4 (𝐴 ∈ dom 𝐺 → (𝐴 ∈ dom (𝐹𝐺) ↔ ∃𝑦𝐴, 𝑦⟩ ∈ (𝐹𝐺)))
2 opelco2g 5821 . . . . . 6 ((𝐴 ∈ dom 𝐺𝑦 ∈ V) → (⟨𝐴, 𝑦⟩ ∈ (𝐹𝐺) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
32elvd 3450 . . . . 5 (𝐴 ∈ dom 𝐺 → (⟨𝐴, 𝑦⟩ ∈ (𝐹𝐺) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
43exbidv 1924 . . . 4 (𝐴 ∈ dom 𝐺 → (∃𝑦𝐴, 𝑦⟩ ∈ (𝐹𝐺) ↔ ∃𝑦𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
51, 4bitrd 278 . . 3 (𝐴 ∈ dom 𝐺 → (𝐴 ∈ dom (𝐹𝐺) ↔ ∃𝑦𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
65adantl 482 . 2 ((Fun 𝐺𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹𝐺) ↔ ∃𝑦𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
7 fvex 6852 . . . 4 (𝐺𝐴) ∈ V
87eldm2 5855 . . 3 ((𝐺𝐴) ∈ dom 𝐹 ↔ ∃𝑦⟨(𝐺𝐴), 𝑦⟩ ∈ 𝐹)
9 opeq1 4828 . . . . . . 7 (𝑥 = (𝐺𝐴) → ⟨𝑥, 𝑦⟩ = ⟨(𝐺𝐴), 𝑦⟩)
109eleq1d 2822 . . . . . 6 (𝑥 = (𝐺𝐴) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨(𝐺𝐴), 𝑦⟩ ∈ 𝐹))
117, 10ceqsexv 3492 . . . . 5 (∃𝑥(𝑥 = (𝐺𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ↔ ⟨(𝐺𝐴), 𝑦⟩ ∈ 𝐹)
12 eqcom 2744 . . . . . . . 8 (𝑥 = (𝐺𝐴) ↔ (𝐺𝐴) = 𝑥)
13 funopfvb 6895 . . . . . . . 8 ((Fun 𝐺𝐴 ∈ dom 𝐺) → ((𝐺𝐴) = 𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐺))
1412, 13bitrid 282 . . . . . . 7 ((Fun 𝐺𝐴 ∈ dom 𝐺) → (𝑥 = (𝐺𝐴) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐺))
1514anbi1d 630 . . . . . 6 ((Fun 𝐺𝐴 ∈ dom 𝐺) → ((𝑥 = (𝐺𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ↔ (⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
1615exbidv 1924 . . . . 5 ((Fun 𝐺𝐴 ∈ dom 𝐺) → (∃𝑥(𝑥 = (𝐺𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
1711, 16bitr3id 284 . . . 4 ((Fun 𝐺𝐴 ∈ dom 𝐺) → (⟨(𝐺𝐴), 𝑦⟩ ∈ 𝐹 ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
1817exbidv 1924 . . 3 ((Fun 𝐺𝐴 ∈ dom 𝐺) → (∃𝑦⟨(𝐺𝐴), 𝑦⟩ ∈ 𝐹 ↔ ∃𝑦𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
198, 18bitrid 282 . 2 ((Fun 𝐺𝐴 ∈ dom 𝐺) → ((𝐺𝐴) ∈ dom 𝐹 ↔ ∃𝑦𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
206, 19bitr4d 281 1 ((Fun 𝐺𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹𝐺) ↔ (𝐺𝐴) ∈ dom 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wex 1781  wcel 2106  Vcvv 3443  cop 4590  dom cdm 5631  ccom 5635  Fun wfun 6487  cfv 6493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6445  df-fun 6495  df-fn 6496  df-fv 6501
This theorem is referenced by:  hoicvr  44683  funressnfv  45171  dmfcoafv  45301  afvco2  45302
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