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Theorem offeq 6035
 Description: Convert an identity of the operation to the analogous identity on the function operation. (Contributed by Jim Kingdon, 26-Nov-2023.)
Hypotheses
Ref Expression
off.1 ((𝜑 ∧ (𝑥𝑆𝑦𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)
off.2 (𝜑𝐹:𝐴𝑆)
off.3 (𝜑𝐺:𝐵𝑇)
off.4 (𝜑𝐴𝑉)
off.5 (𝜑𝐵𝑊)
off.6 (𝐴𝐵) = 𝐶
offeq.4 (𝜑𝐻:𝐶𝑈)
offeq.5 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐷)
offeq.6 ((𝜑𝑥𝐵) → (𝐺𝑥) = 𝐸)
offeq.7 ((𝜑𝑥𝐶) → (𝐷𝑅𝐸) = (𝐻𝑥))
Assertion
Ref Expression
offeq (𝜑 → (𝐹𝑓 𝑅𝐺) = 𝐻)
Distinct variable groups:   𝑦,𝐺,𝑥   𝜑,𝑥,𝑦   𝑥,𝑆,𝑦   𝑥,𝑇,𝑦   𝑥,𝐹,𝑦   𝑥,𝑅,𝑦   𝑥,𝑈,𝑦   𝑥,𝐻   𝑥,𝐺   𝑥,𝐶
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑦)   𝐷(𝑥,𝑦)   𝐸(𝑥,𝑦)   𝐻(𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem offeq
StepHypRef Expression
1 off.1 . . . 4 ((𝜑 ∧ (𝑥𝑆𝑦𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)
2 off.2 . . . 4 (𝜑𝐹:𝐴𝑆)
3 off.3 . . . 4 (𝜑𝐺:𝐵𝑇)
4 off.4 . . . 4 (𝜑𝐴𝑉)
5 off.5 . . . 4 (𝜑𝐵𝑊)
6 off.6 . . . 4 (𝐴𝐵) = 𝐶
71, 2, 3, 4, 5, 6off 6034 . . 3 (𝜑 → (𝐹𝑓 𝑅𝐺):𝐶𝑈)
87ffnd 5313 . 2 (𝜑 → (𝐹𝑓 𝑅𝐺) Fn 𝐶)
9 offeq.4 . . 3 (𝜑𝐻:𝐶𝑈)
109ffnd 5313 . 2 (𝜑𝐻 Fn 𝐶)
112ffnd 5313 . . . 4 (𝜑𝐹 Fn 𝐴)
123ffnd 5313 . . . 4 (𝜑𝐺 Fn 𝐵)
13 offeq.5 . . . 4 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐷)
14 offeq.6 . . . 4 ((𝜑𝑥𝐵) → (𝐺𝑥) = 𝐸)
15 offeq.7 . . . . 5 ((𝜑𝑥𝐶) → (𝐷𝑅𝐸) = (𝐻𝑥))
169ffvelrnda 5595 . . . . 5 ((𝜑𝑥𝐶) → (𝐻𝑥) ∈ 𝑈)
1715, 16eqeltrd 2231 . . . 4 ((𝜑𝑥𝐶) → (𝐷𝑅𝐸) ∈ 𝑈)
1811, 12, 4, 5, 6, 13, 14, 17ofvalg 6031 . . 3 ((𝜑𝑥𝐶) → ((𝐹𝑓 𝑅𝐺)‘𝑥) = (𝐷𝑅𝐸))
1918, 15eqtrd 2187 . 2 ((𝜑𝑥𝐶) → ((𝐹𝑓 𝑅𝐺)‘𝑥) = (𝐻𝑥))
208, 10, 19eqfnfvd 5561 1 (𝜑 → (𝐹𝑓 𝑅𝐺) = 𝐻)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332   ∈ wcel 2125   ∩ cin 3097  ⟶wf 5159  ‘cfv 5163  (class class class)co 5814   ∘𝑓 cof 6020 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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2128  ax-ext 2136  ax-coll 4075  ax-sep 4078  ax-pow 4130  ax-pr 4164  ax-setind 4490 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ne 2325  df-ral 2437  df-rex 2438  df-reu 2439  df-rab 2441  df-v 2711  df-sbc 2934  df-csb 3028  df-dif 3100  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-iun 3847  df-br 3962  df-opab 4022  df-mpt 4023  df-id 4248  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-rn 4590  df-res 4591  df-ima 4592  df-iota 5128  df-fun 5165  df-fn 5166  df-f 5167  df-f1 5168  df-fo 5169  df-f1o 5170  df-fv 5171  df-ov 5817  df-oprab 5818  df-mpo 5819  df-of 6022 This theorem is referenced by:  dviaddf  13016
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