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Theorem afv2co2 43319
Description: Value of a function composition, analogous to fvco2 6754. (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 43314 . . . . . . 7 (𝐺 defAt 𝑋 → {(𝐺''''𝑋)} = (𝐺 “ {𝑋}))
21adantr 481 . . . . . 6 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → {(𝐺''''𝑋)} = (𝐺 “ {𝑋}))
32imaeq2d 5926 . . . . 5 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → (𝐹 “ {(𝐺''''𝑋)}) = (𝐹 “ (𝐺 “ {𝑋})))
4 imaco 6101 . . . . 5 ((𝐹𝐺) “ {𝑋}) = (𝐹 “ (𝐺 “ {𝑋}))
53, 4syl6reqr 2879 . . . 4 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → ((𝐹𝐺) “ {𝑋}) = (𝐹 “ {(𝐺''''𝑋)}))
65eleq2d 2902 . . 3 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → (𝑥 ∈ ((𝐹𝐺) “ {𝑋}) ↔ 𝑥 ∈ (𝐹 “ {(𝐺''''𝑋)})))
76iotabidv 6336 . 2 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → (℩𝑥𝑥 ∈ ((𝐹𝐺) “ {𝑋})) = (℩𝑥𝑥 ∈ (𝐹 “ {(𝐺''''𝑋)})))
8 dfatco 43318 . . 3 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → (𝐹𝐺) defAt 𝑋)
9 dfafv23 43315 . . 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 43315 . . 3 (𝐹 defAt (𝐺''''𝑋) → (𝐹''''(𝐺''''𝑋)) = (℩𝑥𝑥 ∈ (𝐹 “ {(𝐺''''𝑋)})))
1211adantl 482 . 2 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → (𝐹''''(𝐺''''𝑋)) = (℩𝑥𝑥 ∈ (𝐹 “ {(𝐺''''𝑋)})))
137, 10, 123eqtr4d 2870 1 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → ((𝐹𝐺)''''𝑋) = (𝐹''''(𝐺''''𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2106  {csn 4563  cima 5556  ccom 5557  cio 6309   defAt wdfat 43178  ''''cafv2 43270
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 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-dfat 43181  df-afv2 43271
This theorem is referenced by: (None)
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