Theorem afv2co2 47255

Description: Value of a function composition, analogous to fvco2 6958. (Contributed by AV, 8-Sep-2022.)

Assertion

Ref	Expression
afv2co2	⊢ ((𝐺 defAt 𝑋 ∧ 𝐹 defAt (𝐺''''𝑋)) → ((𝐹 ∘ 𝐺)''''𝑋) = (𝐹''''(𝐺''''𝑋)))

Proof of Theorem afv2co2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		imaco 6224	. . . . 5 ⊢ ((𝐹 ∘ 𝐺) “ {𝑋}) = (𝐹 “ (𝐺 “ {𝑋}))
2		dfatsnafv2 47250	. . . . . . 7 ⊢ (𝐺 defAt 𝑋 → {(𝐺''''𝑋)} = (𝐺 “ {𝑋}))
3	2	adantr 480	. . . . . 6 ⊢ ((𝐺 defAt 𝑋 ∧ 𝐹 defAt (𝐺''''𝑋)) → {(𝐺''''𝑋)} = (𝐺 “ {𝑋}))
4	3	imaeq2d 6031	. . . . 5 ⊢ ((𝐺 defAt 𝑋 ∧ 𝐹 defAt (𝐺''''𝑋)) → (𝐹 “ {(𝐺''''𝑋)}) = (𝐹 “ (𝐺 “ {𝑋})))
5	1, 4	eqtr4id 2783	. . . 4 ⊢ ((𝐺 defAt 𝑋 ∧ 𝐹 defAt (𝐺''''𝑋)) → ((𝐹 ∘ 𝐺) “ {𝑋}) = (𝐹 “ {(𝐺''''𝑋)}))
6	5	eleq2d 2814	. . 3 ⊢ ((𝐺 defAt 𝑋 ∧ 𝐹 defAt (𝐺''''𝑋)) → (𝑥 ∈ ((𝐹 ∘ 𝐺) “ {𝑋}) ↔ 𝑥 ∈ (𝐹 “ {(𝐺''''𝑋)})))
7	6	iotabidv 6495	. 2 ⊢ ((𝐺 defAt 𝑋 ∧ 𝐹 defAt (𝐺''''𝑋)) → (℩𝑥𝑥 ∈ ((𝐹 ∘ 𝐺) “ {𝑋})) = (℩𝑥𝑥 ∈ (𝐹 “ {(𝐺''''𝑋)})))
8		dfatco 47254	. . 3 ⊢ ((𝐺 defAt 𝑋 ∧ 𝐹 defAt (𝐺''''𝑋)) → (𝐹 ∘ 𝐺) defAt 𝑋)
9		dfafv23 47251	. . 3 ⊢ ((𝐹 ∘ 𝐺) defAt 𝑋 → ((𝐹 ∘ 𝐺)''''𝑋) = (℩𝑥𝑥 ∈ ((𝐹 ∘ 𝐺) “ {𝑋})))
10	8, 9	syl 17	. 2 ⊢ ((𝐺 defAt 𝑋 ∧ 𝐹 defAt (𝐺''''𝑋)) → ((𝐹 ∘ 𝐺)''''𝑋) = (℩𝑥𝑥 ∈ ((𝐹 ∘ 𝐺) “ {𝑋})))
11		dfafv23 47251	. . 3 ⊢ (𝐹 defAt (𝐺''''𝑋) → (𝐹''''(𝐺''''𝑋)) = (℩𝑥𝑥 ∈ (𝐹 “ {(𝐺''''𝑋)})))
12	11	adantl 481	. 2 ⊢ ((𝐺 defAt 𝑋 ∧ 𝐹 defAt (𝐺''''𝑋)) → (𝐹''''(𝐺''''𝑋)) = (℩𝑥𝑥 ∈ (𝐹 “ {(𝐺''''𝑋)})))
13	7, 10, 12	3eqtr4d 2774	1 ⊢ ((𝐺 defAt 𝑋 ∧ 𝐹 defAt (𝐺''''𝑋)) → ((𝐹 ∘ 𝐺)''''𝑋) = (𝐹''''(𝐺''''𝑋)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {csn 4589   “ cima 5641   ∘ ccom 5642  ℩cio 6462   defAt wdfat 47114  ''''cafv2 47206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-dfat 47117  df-afv2 47207

This theorem is referenced by: (None)