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Theorem ofmpteq 6869
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 1059 . . 3 ((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) → 𝐴𝑉)
2 simpr 477 . . . 4 (((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) ∧ 𝑎𝐴) → 𝑎𝐴)
3 simpl2 1063 . . . . 5 (((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) ∧ 𝑎𝐴) → (𝑥𝐴𝐵) Fn 𝐴)
4 eqid 2621 . . . . . 6 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
54mptfng 5976 . . . . 5 (∀𝑥𝐴 𝐵 ∈ V ↔ (𝑥𝐴𝐵) Fn 𝐴)
63, 5sylibr 224 . . . 4 (((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) ∧ 𝑎𝐴) → ∀𝑥𝐴 𝐵 ∈ V)
7 nfcsb1v 3530 . . . . . 6 𝑥𝑎 / 𝑥𝐵
87nfel1 2775 . . . . 5 𝑥𝑎 / 𝑥𝐵 ∈ V
9 csbeq1a 3523 . . . . . 6 (𝑥 = 𝑎𝐵 = 𝑎 / 𝑥𝐵)
109eleq1d 2683 . . . . 5 (𝑥 = 𝑎 → (𝐵 ∈ V ↔ 𝑎 / 𝑥𝐵 ∈ V))
118, 10rspc 3289 . . . 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 1064 . . . . 5 (((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) ∧ 𝑎𝐴) → (𝑥𝐴𝐶) Fn 𝐴)
14 eqid 2621 . . . . . 6 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
1514mptfng 5976 . . . . 5 (∀𝑥𝐴 𝐶 ∈ V ↔ (𝑥𝐴𝐶) Fn 𝐴)
1613, 15sylibr 224 . . . 4 (((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) ∧ 𝑎𝐴) → ∀𝑥𝐴 𝐶 ∈ V)
17 nfcsb1v 3530 . . . . . 6 𝑥𝑎 / 𝑥𝐶
1817nfel1 2775 . . . . 5 𝑥𝑎 / 𝑥𝐶 ∈ V
19 csbeq1a 3523 . . . . . 6 (𝑥 = 𝑎𝐶 = 𝑎 / 𝑥𝐶)
2019eleq1d 2683 . . . . 5 (𝑥 = 𝑎 → (𝐶 ∈ V ↔ 𝑎 / 𝑥𝐶 ∈ V))
2118, 20rspc 3289 . . . 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 2761 . . . . 5 𝑎𝐵
2423, 7, 9cbvmpt 4709 . . . 4 (𝑥𝐴𝐵) = (𝑎𝐴𝑎 / 𝑥𝐵)
2524a1i 11 . . 3 ((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) → (𝑥𝐴𝐵) = (𝑎𝐴𝑎 / 𝑥𝐵))
26 nfcv 2761 . . . . 5 𝑎𝐶
2726, 17, 19cbvmpt 4709 . . . 4 (𝑥𝐴𝐶) = (𝑎𝐴𝑎 / 𝑥𝐶)
2827a1i 11 . . 3 ((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) → (𝑥𝐴𝐶) = (𝑎𝐴𝑎 / 𝑥𝐶))
291, 12, 22, 25, 28offval2 6867 . 2 ((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) → ((𝑥𝐴𝐵) ∘𝑓 𝑅(𝑥𝐴𝐶)) = (𝑎𝐴 ↦ (𝑎 / 𝑥𝐵𝑅𝑎 / 𝑥𝐶)))
30 nfcv 2761 . . 3 𝑎(𝐵𝑅𝐶)
31 nfcv 2761 . . . 4 𝑥𝑅
327, 31, 17nfov 6630 . . 3 𝑥(𝑎 / 𝑥𝐵𝑅𝑎 / 𝑥𝐶)
339, 19oveq12d 6622 . . 3 (𝑥 = 𝑎 → (𝐵𝑅𝐶) = (𝑎 / 𝑥𝐵𝑅𝑎 / 𝑥𝐶))
3430, 32, 33cbvmpt 4709 . 2 (𝑥𝐴 ↦ (𝐵𝑅𝐶)) = (𝑎𝐴 ↦ (𝑎 / 𝑥𝐵𝑅𝑎 / 𝑥𝐶))
3529, 34syl6eqr 2673 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 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  csb 3514  cmpt 4673   Fn wfn 5842  (class class class)co 6604  𝑓 cof 6848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850
This theorem is referenced by:  mdetrlin  20327  mzpaddmpt  36781  mzpmulmpt  36782  mzpcompact2lem  36791
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