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Theorem dfatco 46263
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 46262 . . . 4 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → ∃!𝑦 𝑋(𝐹𝐺)𝑦)
2 euex 2571 . . . 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 5686 . . . . 5 dom (𝐹𝐺) = {𝑥 ∣ ∃𝑦 𝑥(𝐹𝐺)𝑦}
54eleq2i 2825 . . . 4 (𝑋 ∈ dom (𝐹𝐺) ↔ 𝑋 ∈ {𝑥 ∣ ∃𝑦 𝑥(𝐹𝐺)𝑦})
6 df-dfat 46126 . . . . . . 7 (𝐺 defAt 𝑋 ↔ (𝑋 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝑋})))
76simplbi 498 . . . . . 6 (𝐺 defAt 𝑋𝑋 ∈ dom 𝐺)
87adantr 481 . . . . 5 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → 𝑋 ∈ dom 𝐺)
9 breq1 5151 . . . . . . 7 (𝑥 = 𝑋 → (𝑥(𝐹𝐺)𝑦𝑋(𝐹𝐺)𝑦))
109exbidv 1924 . . . . . 6 (𝑥 = 𝑋 → (∃𝑦 𝑥(𝐹𝐺)𝑦 ↔ ∃𝑦 𝑋(𝐹𝐺)𝑦))
1110elabg 3666 . . . . 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, 12bitrid 282 . . 3 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → (𝑋 ∈ dom (𝐹𝐺) ↔ ∃𝑦 𝑋(𝐹𝐺)𝑦))
143, 13mpbird 256 . 2 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → 𝑋 ∈ dom (𝐹𝐺))
15 dfdfat2 46135 . 2 ((𝐹𝐺) defAt 𝑋 ↔ (𝑋 ∈ dom (𝐹𝐺) ∧ ∃!𝑦 𝑋(𝐹𝐺)𝑦))
1614, 1, 15sylanbrc 583 1 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → (𝐹𝐺) defAt 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wex 1781  wcel 2106  ∃!weu 2562  {cab 2709  {csn 4628   class class class wbr 5148  dom cdm 5676  cres 5678  ccom 5680  Fun wfun 6537   defAt wdfat 46123  ''''cafv2 46215
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-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-res 5688  df-iota 6495  df-fun 6545  df-fn 6546  df-dfat 46126  df-afv2 46216
This theorem is referenced by:  afv2co2  46264
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