Theorem fvco4 5727

Description: Value of a composition. (Contributed by BJ, 7-Jul-2022.)

Assertion

Ref	Expression
fvco4	⊢ (((𝐾:𝐴⟶𝑋 ∧ (𝐻 ∘ 𝐾) = 𝐹) ∧ (𝑢 ∈ 𝐴 ∧ 𝑥 = (𝐾‘𝑢))) → (𝐻‘𝑥) = (𝐹‘𝑢))

Proof of Theorem fvco4

Step	Hyp	Ref	Expression
1		fvco3 5726	. . 3 ⊢ ((𝐾:𝐴⟶𝑋 ∧ 𝑢 ∈ 𝐴) → ((𝐻 ∘ 𝐾)‘𝑢) = (𝐻‘(𝐾‘𝑢)))
2	1	ad2ant2r 509	. 2 ⊢ (((𝐾:𝐴⟶𝑋 ∧ (𝐻 ∘ 𝐾) = 𝐹) ∧ (𝑢 ∈ 𝐴 ∧ 𝑥 = (𝐾‘𝑢))) → ((𝐻 ∘ 𝐾)‘𝑢) = (𝐻‘(𝐾‘𝑢)))
3		simplr 529	. . . 4 ⊢ (((𝐾:𝐴⟶𝑋 ∧ (𝐻 ∘ 𝐾) = 𝐹) ∧ (𝑢 ∈ 𝐴 ∧ 𝑥 = (𝐾‘𝑢))) → (𝐻 ∘ 𝐾) = 𝐹)
4	3	eqcomd 2237	. . 3 ⊢ (((𝐾:𝐴⟶𝑋 ∧ (𝐻 ∘ 𝐾) = 𝐹) ∧ (𝑢 ∈ 𝐴 ∧ 𝑥 = (𝐾‘𝑢))) → 𝐹 = (𝐻 ∘ 𝐾))
5	4	fveq1d 5650	. 2 ⊢ (((𝐾:𝐴⟶𝑋 ∧ (𝐻 ∘ 𝐾) = 𝐹) ∧ (𝑢 ∈ 𝐴 ∧ 𝑥 = (𝐾‘𝑢))) → (𝐹‘𝑢) = ((𝐻 ∘ 𝐾)‘𝑢))
6		fveq2 5648	. . 3 ⊢ (𝑥 = (𝐾‘𝑢) → (𝐻‘𝑥) = (𝐻‘(𝐾‘𝑢)))
7	6	ad2antll 491	. 2 ⊢ (((𝐾:𝐴⟶𝑋 ∧ (𝐻 ∘ 𝐾) = 𝐹) ∧ (𝑢 ∈ 𝐴 ∧ 𝑥 = (𝐾‘𝑢))) → (𝐻‘𝑥) = (𝐻‘(𝐾‘𝑢)))
8	2, 5, 7	3eqtr4rd 2275	1 ⊢ (((𝐾:𝐴⟶𝑋 ∧ (𝐻 ∘ 𝐾) = 𝐹) ∧ (𝑢 ∈ 𝐴 ∧ 𝑥 = (𝐾‘𝑢))) → (𝐻‘𝑥) = (𝐹‘𝑢))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2202   ∘ ccom 4735  ⟶wf 5329  ‘cfv 5333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341

This theorem is referenced by: (None)