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Theorem ofmpteq 7033
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 1128 . . 3 ((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) → 𝐴𝑉)
2 simpr 479 . . . 4 (((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) ∧ 𝑎𝐴) → 𝑎𝐴)
3 simpl2 1206 . . . . 5 (((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) ∧ 𝑎𝐴) → (𝑥𝐴𝐵) Fn 𝐴)
4 eqid 2724 . . . . . 6 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
54mptfng 6132 . . . . 5 (∀𝑥𝐴 𝐵 ∈ V ↔ (𝑥𝐴𝐵) Fn 𝐴)
63, 5sylibr 224 . . . 4 (((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) ∧ 𝑎𝐴) → ∀𝑥𝐴 𝐵 ∈ V)
7 nfcsb1v 3655 . . . . . 6 𝑥𝑎 / 𝑥𝐵
87nfel1 2881 . . . . 5 𝑥𝑎 / 𝑥𝐵 ∈ V
9 csbeq1a 3648 . . . . . 6 (𝑥 = 𝑎𝐵 = 𝑎 / 𝑥𝐵)
109eleq1d 2788 . . . . 5 (𝑥 = 𝑎 → (𝐵 ∈ V ↔ 𝑎 / 𝑥𝐵 ∈ V))
118, 10rspc 3407 . . . 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 1208 . . . . 5 (((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) ∧ 𝑎𝐴) → (𝑥𝐴𝐶) Fn 𝐴)
14 eqid 2724 . . . . . 6 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
1514mptfng 6132 . . . . 5 (∀𝑥𝐴 𝐶 ∈ V ↔ (𝑥𝐴𝐶) Fn 𝐴)
1613, 15sylibr 224 . . . 4 (((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) ∧ 𝑎𝐴) → ∀𝑥𝐴 𝐶 ∈ V)
17 nfcsb1v 3655 . . . . . 6 𝑥𝑎 / 𝑥𝐶
1817nfel1 2881 . . . . 5 𝑥𝑎 / 𝑥𝐶 ∈ V
19 csbeq1a 3648 . . . . . 6 (𝑥 = 𝑎𝐶 = 𝑎 / 𝑥𝐶)
2019eleq1d 2788 . . . . 5 (𝑥 = 𝑎 → (𝐶 ∈ V ↔ 𝑎 / 𝑥𝐶 ∈ V))
2118, 20rspc 3407 . . . 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 2866 . . . . 5 𝑎𝐵
2423, 7, 9cbvmpt 4857 . . . 4 (𝑥𝐴𝐵) = (𝑎𝐴𝑎 / 𝑥𝐵)
2524a1i 11 . . 3 ((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) → (𝑥𝐴𝐵) = (𝑎𝐴𝑎 / 𝑥𝐵))
26 nfcv 2866 . . . . 5 𝑎𝐶
2726, 17, 19cbvmpt 4857 . . . 4 (𝑥𝐴𝐶) = (𝑎𝐴𝑎 / 𝑥𝐶)
2827a1i 11 . . 3 ((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) → (𝑥𝐴𝐶) = (𝑎𝐴𝑎 / 𝑥𝐶))
291, 12, 22, 25, 28offval2 7031 . 2 ((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) → ((𝑥𝐴𝐵) ∘𝑓 𝑅(𝑥𝐴𝐶)) = (𝑎𝐴 ↦ (𝑎 / 𝑥𝐵𝑅𝑎 / 𝑥𝐶)))
30 nfcv 2866 . . 3 𝑎(𝐵𝑅𝐶)
31 nfcv 2866 . . . 4 𝑥𝑅
327, 31, 17nfov 6791 . . 3 𝑥(𝑎 / 𝑥𝐵𝑅𝑎 / 𝑥𝐶)
339, 19oveq12d 6783 . . 3 (𝑥 = 𝑎 → (𝐵𝑅𝐶) = (𝑎 / 𝑥𝐵𝑅𝑎 / 𝑥𝐶))
3430, 32, 33cbvmpt 4857 . 2 (𝑥𝐴 ↦ (𝐵𝑅𝐶)) = (𝑎𝐴 ↦ (𝑎 / 𝑥𝐵𝑅𝑎 / 𝑥𝐶))
3529, 34syl6eqr 2776 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 383  w3a 1072   = wceq 1596  wcel 2103  wral 3014  Vcvv 3304  csb 3639  cmpt 4837   Fn wfn 5996  (class class class)co 6765  𝑓 cof 7012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-of 7014
This theorem is referenced by:  mdetrlin  20531  mzpaddmpt  37723  mzpmulmpt  37724  mzpcompact2lem  37733
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