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Theorem afv2co2 44203
 Description: Value of a function composition, analogous to fvco2 6749. (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 6081 . . . . 5 ((𝐹𝐺) “ {𝑋}) = (𝐹 “ (𝐺 “ {𝑋}))
2 dfatsnafv2 44198 . . . . . . 7 (𝐺 defAt 𝑋 → {(𝐺''''𝑋)} = (𝐺 “ {𝑋}))
32adantr 484 . . . . . 6 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → {(𝐺''''𝑋)} = (𝐺 “ {𝑋}))
43imaeq2d 5901 . . . . 5 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → (𝐹 “ {(𝐺''''𝑋)}) = (𝐹 “ (𝐺 “ {𝑋})))
51, 4eqtr4id 2812 . . . 4 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → ((𝐹𝐺) “ {𝑋}) = (𝐹 “ {(𝐺''''𝑋)}))
65eleq2d 2837 . . 3 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → (𝑥 ∈ ((𝐹𝐺) “ {𝑋}) ↔ 𝑥 ∈ (𝐹 “ {(𝐺''''𝑋)})))
76iotabidv 6319 . 2 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → (℩𝑥𝑥 ∈ ((𝐹𝐺) “ {𝑋})) = (℩𝑥𝑥 ∈ (𝐹 “ {(𝐺''''𝑋)})))
8 dfatco 44202 . . 3 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → (𝐹𝐺) defAt 𝑋)
9 dfafv23 44199 . . 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 44199 . . 3 (𝐹 defAt (𝐺''''𝑋) → (𝐹''''(𝐺''''𝑋)) = (℩𝑥𝑥 ∈ (𝐹 “ {(𝐺''''𝑋)})))
1211adantl 485 . 2 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → (𝐹''''(𝐺''''𝑋)) = (℩𝑥𝑥 ∈ (𝐹 “ {(𝐺''''𝑋)})))
137, 10, 123eqtr4d 2803 1 ((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → ((𝐹𝐺)''''𝑋) = (𝐹''''(𝐺''''𝑋)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {csn 4522   “ cima 5527   ∘ ccom 5528  ℩cio 6292   defAt wdfat 44062  ''''cafv2 44154 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-dfat 44065  df-afv2 44155 This theorem is referenced by: (None)
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