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Theorem dfatco 42004
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 42003 . . . 4 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → ∃!𝑦 𝑋(𝐹𝐺)𝑦)
2 euex 2591 . . . 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 5287 . . . . 5 dom (𝐹𝐺) = {𝑥 ∣ ∃𝑦 𝑥(𝐹𝐺)𝑦}
54eleq2i 2836 . . . 4 (𝑋 ∈ dom (𝐹𝐺) ↔ 𝑋 ∈ {𝑥 ∣ ∃𝑦 𝑥(𝐹𝐺)𝑦})
6 df-dfat 41867 . . . . . . 7 (𝐺 defAt 𝑋 ↔ (𝑋 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝑋})))
76simplbi 491 . . . . . 6 (𝐺 defAt 𝑋𝑋 ∈ dom 𝐺)
87adantr 472 . . . . 5 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → 𝑋 ∈ dom 𝐺)
9 breq1 4812 . . . . . . 7 (𝑥 = 𝑋 → (𝑥(𝐹𝐺)𝑦𝑋(𝐹𝐺)𝑦))
109exbidv 2016 . . . . . 6 (𝑥 = 𝑋 → (∃𝑦 𝑥(𝐹𝐺)𝑦 ↔ ∃𝑦 𝑋(𝐹𝐺)𝑦))
1110elabg 3505 . . . . 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 274 . . 3 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → (𝑋 ∈ dom (𝐹𝐺) ↔ ∃𝑦 𝑋(𝐹𝐺)𝑦))
143, 13mpbird 248 . 2 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → 𝑋 ∈ dom (𝐹𝐺))
15 dfdfat2 41876 . 2 ((𝐹𝐺) defAt 𝑋 ↔ (𝑋 ∈ dom (𝐹𝐺) ∧ ∃!𝑦 𝑋(𝐹𝐺)𝑦))
1614, 1, 15sylanbrc 578 1 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → (𝐹𝐺) defAt 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wex 1874  wcel 2155  ∃!weu 2581  {cab 2751  {csn 4334   class class class wbr 4809  dom cdm 5277  cres 5279  ccom 5281  Fun wfun 6062   defAt wdfat 41864  ''''cafv2 41956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-res 5289  df-iota 6031  df-fun 6070  df-fn 6071  df-dfat 41867  df-afv2 41957
This theorem is referenced by:  afv2co2  42005
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