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Theorem fvco4 5590
Description: Value of a composition. (Contributed by BJ, 7-Jul-2022.)
Assertion
Ref Expression
fvco4 (((𝐾:𝐴𝑋 ∧ (𝐻𝐾) = 𝐹) ∧ (𝑢𝐴𝑥 = (𝐾𝑢))) → (𝐻𝑥) = (𝐹𝑢))

Proof of Theorem fvco4
StepHypRef Expression
1 fvco3 5589 . . 3 ((𝐾:𝐴𝑋𝑢𝐴) → ((𝐻𝐾)‘𝑢) = (𝐻‘(𝐾𝑢)))
21ad2ant2r 509 . 2 (((𝐾:𝐴𝑋 ∧ (𝐻𝐾) = 𝐹) ∧ (𝑢𝐴𝑥 = (𝐾𝑢))) → ((𝐻𝐾)‘𝑢) = (𝐻‘(𝐾𝑢)))
3 simplr 528 . . . 4 (((𝐾:𝐴𝑋 ∧ (𝐻𝐾) = 𝐹) ∧ (𝑢𝐴𝑥 = (𝐾𝑢))) → (𝐻𝐾) = 𝐹)
43eqcomd 2183 . . 3 (((𝐾:𝐴𝑋 ∧ (𝐻𝐾) = 𝐹) ∧ (𝑢𝐴𝑥 = (𝐾𝑢))) → 𝐹 = (𝐻𝐾))
54fveq1d 5519 . 2 (((𝐾:𝐴𝑋 ∧ (𝐻𝐾) = 𝐹) ∧ (𝑢𝐴𝑥 = (𝐾𝑢))) → (𝐹𝑢) = ((𝐻𝐾)‘𝑢))
6 fveq2 5517 . . 3 (𝑥 = (𝐾𝑢) → (𝐻𝑥) = (𝐻‘(𝐾𝑢)))
76ad2antll 491 . 2 (((𝐾:𝐴𝑋 ∧ (𝐻𝐾) = 𝐹) ∧ (𝑢𝐴𝑥 = (𝐾𝑢))) → (𝐻𝑥) = (𝐻‘(𝐾𝑢)))
82, 5, 73eqtr4rd 2221 1 (((𝐾:𝐴𝑋 ∧ (𝐻𝐾) = 𝐹) ∧ (𝑢𝐴𝑥 = (𝐾𝑢))) → (𝐻𝑥) = (𝐹𝑢))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  ccom 4632  wf 5214  cfv 5218
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226
This theorem is referenced by: (None)
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