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Theorem ofmpteq 7720
Description: Value of a pointwise operation on two functions defined using maps-to notation. (Contributed by Stefan O'Rear, 5-Oct-2014.)
Assertion
Ref Expression
ofmpteq ((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) → ((𝑥𝐴𝐵) ∘f 𝑅(𝑥𝐴𝐶)) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑅
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝑉(𝑥)

Proof of Theorem ofmpteq
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 simp1 1137 . . 3 ((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) → 𝐴𝑉)
2 simpr 484 . . . 4 (((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) ∧ 𝑎𝐴) → 𝑎𝐴)
3 simpl2 1193 . . . . 5 (((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) ∧ 𝑎𝐴) → (𝑥𝐴𝐵) Fn 𝐴)
4 eqid 2737 . . . . . 6 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
54mptfng 6707 . . . . 5 (∀𝑥𝐴 𝐵 ∈ V ↔ (𝑥𝐴𝐵) Fn 𝐴)
63, 5sylibr 234 . . . 4 (((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) ∧ 𝑎𝐴) → ∀𝑥𝐴 𝐵 ∈ V)
7 nfcsb1v 3923 . . . . . 6 𝑥𝑎 / 𝑥𝐵
87nfel1 2922 . . . . 5 𝑥𝑎 / 𝑥𝐵 ∈ V
9 csbeq1a 3913 . . . . . 6 (𝑥 = 𝑎𝐵 = 𝑎 / 𝑥𝐵)
109eleq1d 2826 . . . . 5 (𝑥 = 𝑎 → (𝐵 ∈ V ↔ 𝑎 / 𝑥𝐵 ∈ V))
118, 10rspc 3610 . . . 4 (𝑎𝐴 → (∀𝑥𝐴 𝐵 ∈ V → 𝑎 / 𝑥𝐵 ∈ V))
122, 6, 11sylc 65 . . 3 (((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) ∧ 𝑎𝐴) → 𝑎 / 𝑥𝐵 ∈ V)
13 simpl3 1194 . . . . 5 (((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) ∧ 𝑎𝐴) → (𝑥𝐴𝐶) Fn 𝐴)
14 eqid 2737 . . . . . 6 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
1514mptfng 6707 . . . . 5 (∀𝑥𝐴 𝐶 ∈ V ↔ (𝑥𝐴𝐶) Fn 𝐴)
1613, 15sylibr 234 . . . 4 (((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) ∧ 𝑎𝐴) → ∀𝑥𝐴 𝐶 ∈ V)
17 nfcsb1v 3923 . . . . . 6 𝑥𝑎 / 𝑥𝐶
1817nfel1 2922 . . . . 5 𝑥𝑎 / 𝑥𝐶 ∈ V
19 csbeq1a 3913 . . . . . 6 (𝑥 = 𝑎𝐶 = 𝑎 / 𝑥𝐶)
2019eleq1d 2826 . . . . 5 (𝑥 = 𝑎 → (𝐶 ∈ V ↔ 𝑎 / 𝑥𝐶 ∈ V))
2118, 20rspc 3610 . . . 4 (𝑎𝐴 → (∀𝑥𝐴 𝐶 ∈ V → 𝑎 / 𝑥𝐶 ∈ V))
222, 16, 21sylc 65 . . 3 (((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) ∧ 𝑎𝐴) → 𝑎 / 𝑥𝐶 ∈ V)
23 nfcv 2905 . . . . 5 𝑎𝐵
2423, 7, 9cbvmpt 5253 . . . 4 (𝑥𝐴𝐵) = (𝑎𝐴𝑎 / 𝑥𝐵)
2524a1i 11 . . 3 ((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) → (𝑥𝐴𝐵) = (𝑎𝐴𝑎 / 𝑥𝐵))
26 nfcv 2905 . . . . 5 𝑎𝐶
2726, 17, 19cbvmpt 5253 . . . 4 (𝑥𝐴𝐶) = (𝑎𝐴𝑎 / 𝑥𝐶)
2827a1i 11 . . 3 ((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) → (𝑥𝐴𝐶) = (𝑎𝐴𝑎 / 𝑥𝐶))
291, 12, 22, 25, 28offval2 7717 . 2 ((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) → ((𝑥𝐴𝐵) ∘f 𝑅(𝑥𝐴𝐶)) = (𝑎𝐴 ↦ (𝑎 / 𝑥𝐵𝑅𝑎 / 𝑥𝐶)))
30 nfcv 2905 . . 3 𝑎(𝐵𝑅𝐶)
31 nfcv 2905 . . . 4 𝑥𝑅
327, 31, 17nfov 7461 . . 3 𝑥(𝑎 / 𝑥𝐵𝑅𝑎 / 𝑥𝐶)
339, 19oveq12d 7449 . . 3 (𝑥 = 𝑎 → (𝐵𝑅𝐶) = (𝑎 / 𝑥𝐵𝑅𝑎 / 𝑥𝐶))
3430, 32, 33cbvmpt 5253 . 2 (𝑥𝐴 ↦ (𝐵𝑅𝐶)) = (𝑎𝐴 ↦ (𝑎 / 𝑥𝐵𝑅𝑎 / 𝑥𝐶))
3529, 34eqtr4di 2795 1 ((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) → ((𝑥𝐴𝐵) ∘f 𝑅(𝑥𝐴𝐶)) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  Vcvv 3480  csb 3899  cmpt 5225   Fn wfn 6556  (class class class)co 7431  f cof 7695
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697
This theorem is referenced by:  mdetrlin  22608  mzpaddmpt  42752  mzpmulmpt  42753  mzpcompact2lem  42762
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