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Theorem clintopval 41625
Description: The closed (internal binary) operations for a set. (Contributed by AV, 20-Jan-2020.)
Assertion
Ref Expression
clintopval (𝑀𝑉 → ( clIntOp ‘𝑀) = (𝑀𝑚 (𝑀 × 𝑀)))

Proof of Theorem clintopval
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 df-clintop 41621 . . 3 clIntOp = (𝑚 ∈ V ↦ (𝑚 intOp 𝑚))
21a1i 11 . 2 (𝑀𝑉 → clIntOp = (𝑚 ∈ V ↦ (𝑚 intOp 𝑚)))
3 id 22 . . . 4 (𝑚 = 𝑀𝑚 = 𝑀)
43, 3oveq12d 6545 . . 3 (𝑚 = 𝑀 → (𝑚 intOp 𝑚) = (𝑀 intOp 𝑀))
5 intopval 41623 . . . 4 ((𝑀𝑉𝑀𝑉) → (𝑀 intOp 𝑀) = (𝑀𝑚 (𝑀 × 𝑀)))
65anidms 674 . . 3 (𝑀𝑉 → (𝑀 intOp 𝑀) = (𝑀𝑚 (𝑀 × 𝑀)))
74, 6sylan9eqr 2665 . 2 ((𝑀𝑉𝑚 = 𝑀) → (𝑚 intOp 𝑚) = (𝑀𝑚 (𝑀 × 𝑀)))
8 elex 3184 . 2 (𝑀𝑉𝑀 ∈ V)
9 ovex 6555 . . 3 (𝑀𝑚 (𝑀 × 𝑀)) ∈ V
109a1i 11 . 2 (𝑀𝑉 → (𝑀𝑚 (𝑀 × 𝑀)) ∈ V)
112, 7, 8, 10fvmptd 6182 1 (𝑀𝑉 → ( clIntOp ‘𝑀) = (𝑀𝑚 (𝑀 × 𝑀)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  Vcvv 3172  cmpt 4637   × cxp 5026  cfv 5790  (class class class)co 6527  𝑚 cmap 7721   intOp cintop 41617   clIntOp cclintop 41618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-iota 5754  df-fun 5792  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-intop 41620  df-clintop 41621
This theorem is referenced by:  assintopmap  41627  isclintop  41628
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