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Theorem ofco 5966
Description: The composition of a function operation with another function. (Contributed by Mario Carneiro, 19-Dec-2014.)
Hypotheses
Ref Expression
ofco.1 (𝜑𝐹 Fn 𝐴)
ofco.2 (𝜑𝐺 Fn 𝐵)
ofco.3 (𝜑𝐻:𝐷𝐶)
ofco.4 (𝜑𝐴𝑉)
ofco.5 (𝜑𝐵𝑊)
ofco.6 (𝜑𝐷𝑋)
ofco.7 (𝐴𝐵) = 𝐶
Assertion
Ref Expression
ofco (𝜑 → ((𝐹𝑓 𝑅𝐺) ∘ 𝐻) = ((𝐹𝐻) ∘𝑓 𝑅(𝐺𝐻)))

Proof of Theorem ofco
Dummy variables 𝑦 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ofco.3 . . . 4 (𝜑𝐻:𝐷𝐶)
21ffvelrnda 5521 . . 3 ((𝜑𝑥𝐷) → (𝐻𝑥) ∈ 𝐶)
31feqmptd 5440 . . 3 (𝜑𝐻 = (𝑥𝐷 ↦ (𝐻𝑥)))
4 ofco.1 . . . 4 (𝜑𝐹 Fn 𝐴)
5 ofco.2 . . . 4 (𝜑𝐺 Fn 𝐵)
6 ofco.4 . . . 4 (𝜑𝐴𝑉)
7 ofco.5 . . . 4 (𝜑𝐵𝑊)
8 ofco.7 . . . 4 (𝐴𝐵) = 𝐶
9 eqidd 2116 . . . 4 ((𝜑𝑦𝐴) → (𝐹𝑦) = (𝐹𝑦))
10 eqidd 2116 . . . 4 ((𝜑𝑦𝐵) → (𝐺𝑦) = (𝐺𝑦))
114, 5, 6, 7, 8, 9, 10offval 5955 . . 3 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑦𝐶 ↦ ((𝐹𝑦)𝑅(𝐺𝑦))))
12 fveq2 5387 . . . 4 (𝑦 = (𝐻𝑥) → (𝐹𝑦) = (𝐹‘(𝐻𝑥)))
13 fveq2 5387 . . . 4 (𝑦 = (𝐻𝑥) → (𝐺𝑦) = (𝐺‘(𝐻𝑥)))
1412, 13oveq12d 5758 . . 3 (𝑦 = (𝐻𝑥) → ((𝐹𝑦)𝑅(𝐺𝑦)) = ((𝐹‘(𝐻𝑥))𝑅(𝐺‘(𝐻𝑥))))
152, 3, 11, 14fmptco 5552 . 2 (𝜑 → ((𝐹𝑓 𝑅𝐺) ∘ 𝐻) = (𝑥𝐷 ↦ ((𝐹‘(𝐻𝑥))𝑅(𝐺‘(𝐻𝑥)))))
16 inss1 3264 . . . . . 6 (𝐴𝐵) ⊆ 𝐴
178, 16eqsstrri 3098 . . . . 5 𝐶𝐴
18 fss 5252 . . . . 5 ((𝐻:𝐷𝐶𝐶𝐴) → 𝐻:𝐷𝐴)
191, 17, 18sylancl 407 . . . 4 (𝜑𝐻:𝐷𝐴)
20 fnfco 5265 . . . 4 ((𝐹 Fn 𝐴𝐻:𝐷𝐴) → (𝐹𝐻) Fn 𝐷)
214, 19, 20syl2anc 406 . . 3 (𝜑 → (𝐹𝐻) Fn 𝐷)
22 inss2 3265 . . . . . 6 (𝐴𝐵) ⊆ 𝐵
238, 22eqsstrri 3098 . . . . 5 𝐶𝐵
24 fss 5252 . . . . 5 ((𝐻:𝐷𝐶𝐶𝐵) → 𝐻:𝐷𝐵)
251, 23, 24sylancl 407 . . . 4 (𝜑𝐻:𝐷𝐵)
26 fnfco 5265 . . . 4 ((𝐺 Fn 𝐵𝐻:𝐷𝐵) → (𝐺𝐻) Fn 𝐷)
275, 25, 26syl2anc 406 . . 3 (𝜑 → (𝐺𝐻) Fn 𝐷)
28 ofco.6 . . 3 (𝜑𝐷𝑋)
29 inidm 3253 . . 3 (𝐷𝐷) = 𝐷
30 ffn 5240 . . . . 5 (𝐻:𝐷𝐶𝐻 Fn 𝐷)
311, 30syl 14 . . . 4 (𝜑𝐻 Fn 𝐷)
32 fvco2 5456 . . . 4 ((𝐻 Fn 𝐷𝑥𝐷) → ((𝐹𝐻)‘𝑥) = (𝐹‘(𝐻𝑥)))
3331, 32sylan 279 . . 3 ((𝜑𝑥𝐷) → ((𝐹𝐻)‘𝑥) = (𝐹‘(𝐻𝑥)))
34 fvco2 5456 . . . 4 ((𝐻 Fn 𝐷𝑥𝐷) → ((𝐺𝐻)‘𝑥) = (𝐺‘(𝐻𝑥)))
3531, 34sylan 279 . . 3 ((𝜑𝑥𝐷) → ((𝐺𝐻)‘𝑥) = (𝐺‘(𝐻𝑥)))
3621, 27, 28, 28, 29, 33, 35offval 5955 . 2 (𝜑 → ((𝐹𝐻) ∘𝑓 𝑅(𝐺𝐻)) = (𝑥𝐷 ↦ ((𝐹‘(𝐻𝑥))𝑅(𝐺‘(𝐻𝑥)))))
3715, 36eqtr4d 2151 1 (𝜑 → ((𝐹𝑓 𝑅𝐺) ∘ 𝐻) = ((𝐹𝐻) ∘𝑓 𝑅(𝐺𝐻)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1314  wcel 1463  cin 3038  wss 3039  cmpt 3957  ccom 4511   Fn wfn 5086  wf 5087  cfv 5091  (class class class)co 5740  𝑓 cof 5946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-setind 4420
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-of 5948
This theorem is referenced by: (None)
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