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Theorem dfatco 44449
Description: The predicate "defined at" for a function composition. (Contributed by AV, 8-Sep-2022.)
Assertion
Ref Expression
dfatco ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → (𝐹𝐺) defAt 𝑋)

Proof of Theorem dfatco
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfatcolem 44448 . . . 4 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → ∃!𝑦 𝑋(𝐹𝐺)𝑦)
2 euex 2577 . . . 4 (∃!𝑦 𝑋(𝐹𝐺)𝑦 → ∃𝑦 𝑋(𝐹𝐺)𝑦)
31, 2syl 17 . . 3 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → ∃𝑦 𝑋(𝐹𝐺)𝑦)
4 df-dm 5576 . . . . 5 dom (𝐹𝐺) = {𝑥 ∣ ∃𝑦 𝑥(𝐹𝐺)𝑦}
54eleq2i 2830 . . . 4 (𝑋 ∈ dom (𝐹𝐺) ↔ 𝑋 ∈ {𝑥 ∣ ∃𝑦 𝑥(𝐹𝐺)𝑦})
6 df-dfat 44312 . . . . . . 7 (𝐺 defAt 𝑋 ↔ (𝑋 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝑋})))
76simplbi 501 . . . . . 6 (𝐺 defAt 𝑋𝑋 ∈ dom 𝐺)
87adantr 484 . . . . 5 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → 𝑋 ∈ dom 𝐺)
9 breq1 5071 . . . . . . 7 (𝑥 = 𝑋 → (𝑥(𝐹𝐺)𝑦𝑋(𝐹𝐺)𝑦))
109exbidv 1929 . . . . . 6 (𝑥 = 𝑋 → (∃𝑦 𝑥(𝐹𝐺)𝑦 ↔ ∃𝑦 𝑋(𝐹𝐺)𝑦))
1110elabg 3598 . . . . 5 (𝑋 ∈ dom 𝐺 → (𝑋 ∈ {𝑥 ∣ ∃𝑦 𝑥(𝐹𝐺)𝑦} ↔ ∃𝑦 𝑋(𝐹𝐺)𝑦))
128, 11syl 17 . . . 4 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → (𝑋 ∈ {𝑥 ∣ ∃𝑦 𝑥(𝐹𝐺)𝑦} ↔ ∃𝑦 𝑋(𝐹𝐺)𝑦))
135, 12syl5bb 286 . . 3 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → (𝑋 ∈ dom (𝐹𝐺) ↔ ∃𝑦 𝑋(𝐹𝐺)𝑦))
143, 13mpbird 260 . 2 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → 𝑋 ∈ dom (𝐹𝐺))
15 dfdfat2 44321 . 2 ((𝐹𝐺) defAt 𝑋 ↔ (𝑋 ∈ dom (𝐹𝐺) ∧ ∃!𝑦 𝑋(𝐹𝐺)𝑦))
1614, 1, 15sylanbrc 586 1 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → (𝐹𝐺) defAt 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wex 1787  wcel 2111  ∃!weu 2568  {cab 2715  {csn 4556   class class class wbr 5068  dom cdm 5566  cres 5568  ccom 5570  Fun wfun 6392   defAt wdfat 44309  ''''cafv2 44401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-sep 5207  ax-nul 5214  ax-pow 5273  ax-pr 5337
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3423  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4253  df-if 4455  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4835  df-br 5069  df-opab 5131  df-id 5470  df-xp 5572  df-rel 5573  df-cnv 5574  df-co 5575  df-dm 5576  df-res 5578  df-iota 6356  df-fun 6400  df-fn 6401  df-dfat 44312  df-afv2 44402
This theorem is referenced by:  afv2co2  44450
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