MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ofmpteq Structured version   Visualization version   GIF version

Theorem ofmpteq 7063
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 1130 . . 3 ((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) → 𝐴𝑉)
2 simpr 471 . . . 4 (((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) ∧ 𝑎𝐴) → 𝑎𝐴)
3 simpl2 1229 . . . . 5 (((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) ∧ 𝑎𝐴) → (𝑥𝐴𝐵) Fn 𝐴)
4 eqid 2771 . . . . . 6 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
54mptfng 6159 . . . . 5 (∀𝑥𝐴 𝐵 ∈ V ↔ (𝑥𝐴𝐵) Fn 𝐴)
63, 5sylibr 224 . . . 4 (((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) ∧ 𝑎𝐴) → ∀𝑥𝐴 𝐵 ∈ V)
7 nfcsb1v 3698 . . . . . 6 𝑥𝑎 / 𝑥𝐵
87nfel1 2928 . . . . 5 𝑥𝑎 / 𝑥𝐵 ∈ V
9 csbeq1a 3691 . . . . . 6 (𝑥 = 𝑎𝐵 = 𝑎 / 𝑥𝐵)
109eleq1d 2835 . . . . 5 (𝑥 = 𝑎 → (𝐵 ∈ V ↔ 𝑎 / 𝑥𝐵 ∈ V))
118, 10rspc 3454 . . . 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 1231 . . . . 5 (((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) ∧ 𝑎𝐴) → (𝑥𝐴𝐶) Fn 𝐴)
14 eqid 2771 . . . . . 6 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
1514mptfng 6159 . . . . 5 (∀𝑥𝐴 𝐶 ∈ V ↔ (𝑥𝐴𝐶) Fn 𝐴)
1613, 15sylibr 224 . . . 4 (((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) ∧ 𝑎𝐴) → ∀𝑥𝐴 𝐶 ∈ V)
17 nfcsb1v 3698 . . . . . 6 𝑥𝑎 / 𝑥𝐶
1817nfel1 2928 . . . . 5 𝑥𝑎 / 𝑥𝐶 ∈ V
19 csbeq1a 3691 . . . . . 6 (𝑥 = 𝑎𝐶 = 𝑎 / 𝑥𝐶)
2019eleq1d 2835 . . . . 5 (𝑥 = 𝑎 → (𝐶 ∈ V ↔ 𝑎 / 𝑥𝐶 ∈ V))
2118, 20rspc 3454 . . . 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 2913 . . . . 5 𝑎𝐵
2423, 7, 9cbvmpt 4883 . . . 4 (𝑥𝐴𝐵) = (𝑎𝐴𝑎 / 𝑥𝐵)
2524a1i 11 . . 3 ((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) → (𝑥𝐴𝐵) = (𝑎𝐴𝑎 / 𝑥𝐵))
26 nfcv 2913 . . . . 5 𝑎𝐶
2726, 17, 19cbvmpt 4883 . . . 4 (𝑥𝐴𝐶) = (𝑎𝐴𝑎 / 𝑥𝐶)
2827a1i 11 . . 3 ((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) → (𝑥𝐴𝐶) = (𝑎𝐴𝑎 / 𝑥𝐶))
291, 12, 22, 25, 28offval2 7061 . 2 ((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) → ((𝑥𝐴𝐵) ∘𝑓 𝑅(𝑥𝐴𝐶)) = (𝑎𝐴 ↦ (𝑎 / 𝑥𝐵𝑅𝑎 / 𝑥𝐶)))
30 nfcv 2913 . . 3 𝑎(𝐵𝑅𝐶)
31 nfcv 2913 . . . 4 𝑥𝑅
327, 31, 17nfov 6821 . . 3 𝑥(𝑎 / 𝑥𝐵𝑅𝑎 / 𝑥𝐶)
339, 19oveq12d 6811 . . 3 (𝑥 = 𝑎 → (𝐵𝑅𝐶) = (𝑎 / 𝑥𝐵𝑅𝑎 / 𝑥𝐶))
3430, 32, 33cbvmpt 4883 . 2 (𝑥𝐴 ↦ (𝐵𝑅𝐶)) = (𝑎𝐴 ↦ (𝑎 / 𝑥𝐵𝑅𝑎 / 𝑥𝐶))
3529, 34syl6eqr 2823 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 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  Vcvv 3351  csb 3682  cmpt 4863   Fn wfn 6026  (class class class)co 6793  𝑓 cof 7042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-of 7044
This theorem is referenced by:  mdetrlin  20626  mzpaddmpt  37830  mzpmulmpt  37831  mzpcompact2lem  37840
  Copyright terms: Public domain W3C validator