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Theorem dmfco 6519
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 5552 . . . 4 (𝐴 ∈ dom 𝐺 → (𝐴 ∈ dom (𝐹𝐺) ↔ ∃𝑦𝐴, 𝑦⟩ ∈ (𝐹𝐺)))
2 opelco2g 5522 . . . . . 6 ((𝐴 ∈ dom 𝐺𝑦 ∈ V) → (⟨𝐴, 𝑦⟩ ∈ (𝐹𝐺) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
32elvd 3419 . . . . 5 (𝐴 ∈ dom 𝐺 → (⟨𝐴, 𝑦⟩ ∈ (𝐹𝐺) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
43exbidv 2020 . . . 4 (𝐴 ∈ dom 𝐺 → (∃𝑦𝐴, 𝑦⟩ ∈ (𝐹𝐺) ↔ ∃𝑦𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
51, 4bitrd 271 . . 3 (𝐴 ∈ dom 𝐺 → (𝐴 ∈ dom (𝐹𝐺) ↔ ∃𝑦𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
65adantl 475 . 2 ((Fun 𝐺𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹𝐺) ↔ ∃𝑦𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
7 fvex 6446 . . . 4 (𝐺𝐴) ∈ V
87eldm2 5554 . . 3 ((𝐺𝐴) ∈ dom 𝐹 ↔ ∃𝑦⟨(𝐺𝐴), 𝑦⟩ ∈ 𝐹)
9 opeq1 4623 . . . . . . 7 (𝑥 = (𝐺𝐴) → ⟨𝑥, 𝑦⟩ = ⟨(𝐺𝐴), 𝑦⟩)
109eleq1d 2891 . . . . . 6 (𝑥 = (𝐺𝐴) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨(𝐺𝐴), 𝑦⟩ ∈ 𝐹))
117, 10ceqsexv 3459 . . . . 5 (∃𝑥(𝑥 = (𝐺𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ↔ ⟨(𝐺𝐴), 𝑦⟩ ∈ 𝐹)
12 eqcom 2832 . . . . . . . 8 (𝑥 = (𝐺𝐴) ↔ (𝐺𝐴) = 𝑥)
13 funopfvb 6485 . . . . . . . 8 ((Fun 𝐺𝐴 ∈ dom 𝐺) → ((𝐺𝐴) = 𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐺))
1412, 13syl5bb 275 . . . . . . 7 ((Fun 𝐺𝐴 ∈ dom 𝐺) → (𝑥 = (𝐺𝐴) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐺))
1514anbi1d 623 . . . . . 6 ((Fun 𝐺𝐴 ∈ dom 𝐺) → ((𝑥 = (𝐺𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ↔ (⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
1615exbidv 2020 . . . . 5 ((Fun 𝐺𝐴 ∈ dom 𝐺) → (∃𝑥(𝑥 = (𝐺𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
1711, 16syl5bbr 277 . . . 4 ((Fun 𝐺𝐴 ∈ dom 𝐺) → (⟨(𝐺𝐴), 𝑦⟩ ∈ 𝐹 ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
1817exbidv 2020 . . 3 ((Fun 𝐺𝐴 ∈ dom 𝐺) → (∃𝑦⟨(𝐺𝐴), 𝑦⟩ ∈ 𝐹 ↔ ∃𝑦𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
198, 18syl5bb 275 . 2 ((Fun 𝐺𝐴 ∈ dom 𝐺) → ((𝐺𝐴) ∈ dom 𝐹 ↔ ∃𝑦𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
206, 19bitr4d 274 1 ((Fun 𝐺𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹𝐺) ↔ (𝐺𝐴) ∈ dom 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1656  wex 1878  wcel 2164  Vcvv 3414  cop 4403  dom cdm 5342  ccom 5346  Fun wfun 6117  cfv 6123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-iota 6086  df-fun 6125  df-fn 6126  df-fv 6131
This theorem is referenced by:  hoicvr  41549  funressnfv  41967  dmfcoafv  42070  afvco2  42071
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