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Theorem afv2co2 46264
Description: Value of a function composition, analogous to fvco2 6988. (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 imaco 6250 . . . . 5 ((𝐹𝐺) “ {𝑋}) = (𝐹 “ (𝐺 “ {𝑋}))
2 dfatsnafv2 46259 . . . . . . 7 (𝐺 defAt 𝑋 → {(𝐺''''𝑋)} = (𝐺 “ {𝑋}))
32adantr 481 . . . . . 6 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → {(𝐺''''𝑋)} = (𝐺 “ {𝑋}))
43imaeq2d 6059 . . . . 5 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → (𝐹 “ {(𝐺''''𝑋)}) = (𝐹 “ (𝐺 “ {𝑋})))
51, 4eqtr4id 2791 . . . 4 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → ((𝐹𝐺) “ {𝑋}) = (𝐹 “ {(𝐺''''𝑋)}))
65eleq2d 2819 . . 3 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → (𝑥 ∈ ((𝐹𝐺) “ {𝑋}) ↔ 𝑥 ∈ (𝐹 “ {(𝐺''''𝑋)})))
76iotabidv 6527 . 2 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → (℩𝑥𝑥 ∈ ((𝐹𝐺) “ {𝑋})) = (℩𝑥𝑥 ∈ (𝐹 “ {(𝐺''''𝑋)})))
8 dfatco 46263 . . 3 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → (𝐹𝐺) defAt 𝑋)
9 dfafv23 46260 . . 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 46260 . . 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 2782 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 1541  wcel 2106  {csn 4628  cima 5679  ccom 5680  cio 6493   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-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-dfat 46126  df-afv2 46216
This theorem is referenced by: (None)
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