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Theorem afv2co2 42099
Description: Value of a function composition, analogous to fvco2 6496. (Contributed by AV, 8-Sep-2022.)
Assertion
Ref Expression
afv2co2 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → ((𝐹𝐺)''''𝑋) = (𝐹''''(𝐺''''𝑋)))

Proof of Theorem afv2co2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dfatsnafv2 42094 . . . . . . 7 (𝐺 defAt 𝑋 → {(𝐺''''𝑋)} = (𝐺 “ {𝑋}))
21adantr 473 . . . . . 6 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → {(𝐺''''𝑋)} = (𝐺 “ {𝑋}))
32imaeq2d 5681 . . . . 5 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → (𝐹 “ {(𝐺''''𝑋)}) = (𝐹 “ (𝐺 “ {𝑋})))
4 imaco 5857 . . . . 5 ((𝐹𝐺) “ {𝑋}) = (𝐹 “ (𝐺 “ {𝑋}))
53, 4syl6reqr 2850 . . . 4 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → ((𝐹𝐺) “ {𝑋}) = (𝐹 “ {(𝐺''''𝑋)}))
65eleq2d 2862 . . 3 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → (𝑥 ∈ ((𝐹𝐺) “ {𝑋}) ↔ 𝑥 ∈ (𝐹 “ {(𝐺''''𝑋)})))
76iotabidv 6083 . 2 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → (℩𝑥𝑥 ∈ ((𝐹𝐺) “ {𝑋})) = (℩𝑥𝑥 ∈ (𝐹 “ {(𝐺''''𝑋)})))
8 dfatco 42098 . . 3 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → (𝐹𝐺) defAt 𝑋)
9 dfafv23 42095 . . 3 ((𝐹𝐺) defAt 𝑋 → ((𝐹𝐺)''''𝑋) = (℩𝑥𝑥 ∈ ((𝐹𝐺) “ {𝑋})))
108, 9syl 17 . 2 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → ((𝐹𝐺)''''𝑋) = (℩𝑥𝑥 ∈ ((𝐹𝐺) “ {𝑋})))
11 dfafv23 42095 . . 3 (𝐹 defAt (𝐺''''𝑋) → (𝐹''''(𝐺''''𝑋)) = (℩𝑥𝑥 ∈ (𝐹 “ {(𝐺''''𝑋)})))
1211adantl 474 . 2 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → (𝐹''''(𝐺''''𝑋)) = (℩𝑥𝑥 ∈ (𝐹 “ {(𝐺''''𝑋)})))
137, 10, 123eqtr4d 2841 1 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → ((𝐹𝐺)''''𝑋) = (𝐹''''(𝐺''''𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  {csn 4366  cima 5313  ccom 5314  cio 6060   defAt wdfat 41958  ''''cafv2 42050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-br 4842  df-opab 4904  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-dfat 41961  df-afv2 42051
This theorem is referenced by: (None)
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