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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 1135 . . 3 ((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) → 𝐴𝑉)
2 simpr 484 . . . 4 (((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) ∧ 𝑎𝐴) → 𝑎𝐴)
3 simpl2 1191 . . . . 5 (((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) ∧ 𝑎𝐴) → (𝑥𝐴𝐵) Fn 𝐴)
4 eqid 2735 . . . . . 6 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
54mptfng 6708 . . . . 5 (∀𝑥𝐴 𝐵 ∈ V ↔ (𝑥𝐴𝐵) Fn 𝐴)
63, 5sylibr 234 . . . 4 (((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) ∧ 𝑎𝐴) → ∀𝑥𝐴 𝐵 ∈ V)
7 nfcsb1v 3933 . . . . . 6 𝑥𝑎 / 𝑥𝐵
87nfel1 2920 . . . . 5 𝑥𝑎 / 𝑥𝐵 ∈ V
9 csbeq1a 3922 . . . . . 6 (𝑥 = 𝑎𝐵 = 𝑎 / 𝑥𝐵)
109eleq1d 2824 . . . . 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 1192 . . . . 5 (((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) ∧ 𝑎𝐴) → (𝑥𝐴𝐶) Fn 𝐴)
14 eqid 2735 . . . . . 6 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
1514mptfng 6708 . . . . 5 (∀𝑥𝐴 𝐶 ∈ V ↔ (𝑥𝐴𝐶) Fn 𝐴)
1613, 15sylibr 234 . . . 4 (((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) ∧ 𝑎𝐴) → ∀𝑥𝐴 𝐶 ∈ V)
17 nfcsb1v 3933 . . . . . 6 𝑥𝑎 / 𝑥𝐶
1817nfel1 2920 . . . . 5 𝑥𝑎 / 𝑥𝐶 ∈ V
19 csbeq1a 3922 . . . . . 6 (𝑥 = 𝑎𝐶 = 𝑎 / 𝑥𝐶)
2019eleq1d 2824 . . . . 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 2903 . . . . 5 𝑎𝐵
2423, 7, 9cbvmpt 5259 . . . 4 (𝑥𝐴𝐵) = (𝑎𝐴𝑎 / 𝑥𝐵)
2524a1i 11 . . 3 ((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) → (𝑥𝐴𝐵) = (𝑎𝐴𝑎 / 𝑥𝐵))
26 nfcv 2903 . . . . 5 𝑎𝐶
2726, 17, 19cbvmpt 5259 . . . 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 2903 . . 3 𝑎(𝐵𝑅𝐶)
31 nfcv 2903 . . . 4 𝑥𝑅
327, 31, 17nfov 7461 . . 3 𝑥(𝑎 / 𝑥𝐵𝑅𝑎 / 𝑥𝐶)
339, 19oveq12d 7449 . . 3 (𝑥 = 𝑎 → (𝐵𝑅𝐶) = (𝑎 / 𝑥𝐵𝑅𝑎 / 𝑥𝐶))
3430, 32, 33cbvmpt 5259 . 2 (𝑥𝐴 ↦ (𝐵𝑅𝐶)) = (𝑎𝐴 ↦ (𝑎 / 𝑥𝐵𝑅𝑎 / 𝑥𝐶))
3529, 34eqtr4di 2793 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 1086   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  csb 3908  cmpt 5231   Fn wfn 6558  (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 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697
This theorem is referenced by:  mdetrlin  22624  mzpaddmpt  42729  mzpmulmpt  42730  mzpcompact2lem  42739
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