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Theorem mptelpm 44579
Description: A function in maps-to notation is a partial map . (Contributed by Glauco Siliprandi, 5-Apr-2020.)
Hypotheses
Ref Expression
mptelpm.b ((𝜑𝑥𝐴) → 𝐵𝐶)
mptelpm.a (𝜑𝐴𝐷)
mptelpm.c (𝜑𝐶𝑉)
mptelpm.d (𝜑𝐷𝑊)
Assertion
Ref Expression
mptelpm (𝜑 → (𝑥𝐴𝐵) ∈ (𝐶pm 𝐷))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝜑,𝑥
Allowed substitution hints:   𝐵(𝑥)   𝐷(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem mptelpm
StepHypRef Expression
1 mptelpm.b . . . . 5 ((𝜑𝑥𝐴) → 𝐵𝐶)
21fmpttd 7130 . . . 4 (𝜑 → (𝑥𝐴𝐵):𝐴𝐶)
3 eqid 2728 . . . . . . 7 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
43, 1dmmptd 6705 . . . . . 6 (𝜑 → dom (𝑥𝐴𝐵) = 𝐴)
54eqcomd 2734 . . . . 5 (𝜑𝐴 = dom (𝑥𝐴𝐵))
65feq2d 6713 . . . 4 (𝜑 → ((𝑥𝐴𝐵):𝐴𝐶 ↔ (𝑥𝐴𝐵):dom (𝑥𝐴𝐵)⟶𝐶))
72, 6mpbid 231 . . 3 (𝜑 → (𝑥𝐴𝐵):dom (𝑥𝐴𝐵)⟶𝐶)
8 mptelpm.a . . . 4 (𝜑𝐴𝐷)
94, 8eqsstrd 4020 . . 3 (𝜑 → dom (𝑥𝐴𝐵) ⊆ 𝐷)
107, 9jca 510 . 2 (𝜑 → ((𝑥𝐴𝐵):dom (𝑥𝐴𝐵)⟶𝐶 ∧ dom (𝑥𝐴𝐵) ⊆ 𝐷))
11 mptelpm.c . . 3 (𝜑𝐶𝑉)
12 mptelpm.d . . 3 (𝜑𝐷𝑊)
13 elpm2g 8869 . . 3 ((𝐶𝑉𝐷𝑊) → ((𝑥𝐴𝐵) ∈ (𝐶pm 𝐷) ↔ ((𝑥𝐴𝐵):dom (𝑥𝐴𝐵)⟶𝐶 ∧ dom (𝑥𝐴𝐵) ⊆ 𝐷)))
1411, 12, 13syl2anc 582 . 2 (𝜑 → ((𝑥𝐴𝐵) ∈ (𝐶pm 𝐷) ↔ ((𝑥𝐴𝐵):dom (𝑥𝐴𝐵)⟶𝐶 ∧ dom (𝑥𝐴𝐵) ⊆ 𝐷)))
1510, 14mpbird 256 1 (𝜑 → (𝑥𝐴𝐵) ∈ (𝐶pm 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wcel 2098  wss 3949  cmpt 5235  dom cdm 5682  wf 6549  (class class class)co 7426  pm cpm 8852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-pm 8854
This theorem is referenced by:  dvnmptconst  45358  dvnmul  45360
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