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Theorem dfatco 46774
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 46773 . . . 4 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → ∃!𝑦 𝑋(𝐹𝐺)𝑦)
2 euex 2565 . . . 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 5688 . . . . 5 dom (𝐹𝐺) = {𝑥 ∣ ∃𝑦 𝑥(𝐹𝐺)𝑦}
54eleq2i 2817 . . . 4 (𝑋 ∈ dom (𝐹𝐺) ↔ 𝑋 ∈ {𝑥 ∣ ∃𝑦 𝑥(𝐹𝐺)𝑦})
6 df-dfat 46637 . . . . . . 7 (𝐺 defAt 𝑋 ↔ (𝑋 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝑋})))
76simplbi 496 . . . . . 6 (𝐺 defAt 𝑋𝑋 ∈ dom 𝐺)
87adantr 479 . . . . 5 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → 𝑋 ∈ dom 𝐺)
9 breq1 5152 . . . . . . 7 (𝑥 = 𝑋 → (𝑥(𝐹𝐺)𝑦𝑋(𝐹𝐺)𝑦))
109exbidv 1916 . . . . . 6 (𝑥 = 𝑋 → (∃𝑦 𝑥(𝐹𝐺)𝑦 ↔ ∃𝑦 𝑋(𝐹𝐺)𝑦))
1110elabg 3662 . . . . 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 46646 . 2 ((𝐹𝐺) defAt 𝑋 ↔ (𝑋 ∈ dom (𝐹𝐺) ∧ ∃!𝑦 𝑋(𝐹𝐺)𝑦))
1614, 1, 15sylanbrc 581 1 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → (𝐹𝐺) defAt 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wex 1773  wcel 2098  ∃!weu 2556  {cab 2702  {csn 4630   class class class wbr 5149  dom cdm 5678  cres 5680  ccom 5682  Fun wfun 6543   defAt wdfat 46634  ''''cafv2 46726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-res 5690  df-iota 6501  df-fun 6551  df-fn 6552  df-dfat 46637  df-afv2 46727
This theorem is referenced by:  afv2co2  46775
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