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Theorem mpt2curryvald 7341
Description: The value of a curried operation given in maps-to notation is a function over the second argument of the original operation. (Contributed by AV, 27-Oct-2019.)
Hypotheses
Ref Expression
mpt2curryd.f 𝐹 = (𝑥𝑋, 𝑦𝑌𝐶)
mpt2curryd.c (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐶𝑉)
mpt2curryd.n (𝜑𝑌 ≠ ∅)
mpt2curryvald.y (𝜑𝑌𝑊)
mpt2curryvald.a (𝜑𝐴𝑋)
Assertion
Ref Expression
mpt2curryvald (𝜑 → (curry 𝐹𝐴) = (𝑦𝑌𝐴 / 𝑥𝐶))
Distinct variable groups:   𝑥,𝐹,𝑦   𝑥,𝑉,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝜑,𝑥,𝑦   𝑥,𝐴,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem mpt2curryvald
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 mpt2curryd.f . . . 4 𝐹 = (𝑥𝑋, 𝑦𝑌𝐶)
2 mpt2curryd.c . . . 4 (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐶𝑉)
3 mpt2curryd.n . . . 4 (𝜑𝑌 ≠ ∅)
41, 2, 3mpt2curryd 7340 . . 3 (𝜑 → curry 𝐹 = (𝑥𝑋 ↦ (𝑦𝑌𝐶)))
5 nfcv 2761 . . . 4 𝑎(𝑦𝑌𝐶)
6 nfcv 2761 . . . . 5 𝑥𝑌
7 nfcsb1v 3530 . . . . 5 𝑥𝑎 / 𝑥𝐶
86, 7nfmpt 4706 . . . 4 𝑥(𝑦𝑌𝑎 / 𝑥𝐶)
9 csbeq1a 3523 . . . . 5 (𝑥 = 𝑎𝐶 = 𝑎 / 𝑥𝐶)
109mpteq2dv 4705 . . . 4 (𝑥 = 𝑎 → (𝑦𝑌𝐶) = (𝑦𝑌𝑎 / 𝑥𝐶))
115, 8, 10cbvmpt 4709 . . 3 (𝑥𝑋 ↦ (𝑦𝑌𝐶)) = (𝑎𝑋 ↦ (𝑦𝑌𝑎 / 𝑥𝐶))
124, 11syl6eq 2671 . 2 (𝜑 → curry 𝐹 = (𝑎𝑋 ↦ (𝑦𝑌𝑎 / 𝑥𝐶)))
13 csbeq1 3517 . . . 4 (𝑎 = 𝐴𝑎 / 𝑥𝐶 = 𝐴 / 𝑥𝐶)
1413adantl 482 . . 3 ((𝜑𝑎 = 𝐴) → 𝑎 / 𝑥𝐶 = 𝐴 / 𝑥𝐶)
1514mpteq2dv 4705 . 2 ((𝜑𝑎 = 𝐴) → (𝑦𝑌𝑎 / 𝑥𝐶) = (𝑦𝑌𝐴 / 𝑥𝐶))
16 mpt2curryvald.a . 2 (𝜑𝐴𝑋)
17 mpt2curryvald.y . . 3 (𝜑𝑌𝑊)
18 mptexg 6438 . . 3 (𝑌𝑊 → (𝑦𝑌𝐴 / 𝑥𝐶) ∈ V)
1917, 18syl 17 . 2 (𝜑 → (𝑦𝑌𝐴 / 𝑥𝐶) ∈ V)
2012, 15, 16, 19fvmptd 6245 1 (𝜑 → (curry 𝐹𝐴) = (𝑦𝑌𝐴 / 𝑥𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wne 2790  wral 2907  Vcvv 3186  csb 3514  c0 3891  cmpt 4673  cfv 5847  cmpt2 6606  curry ccur 7336
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  ax-un 6902
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-1st 7113  df-2nd 7114  df-cur 7338
This theorem is referenced by:  fvmpt2curryd  7342  pmatcollpw3lem  20507  logbmpt  24426
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