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Theorem mptelpm 41639
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 6860 . . . 4 (𝜑 → (𝑥𝐴𝐵):𝐴𝐶)
3 eqid 2824 . . . . . . 7 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
43, 1dmmptd 6474 . . . . . 6 (𝜑 → dom (𝑥𝐴𝐵) = 𝐴)
54eqcomd 2830 . . . . 5 (𝜑𝐴 = dom (𝑥𝐴𝐵))
65feq2d 6481 . . . 4 (𝜑 → ((𝑥𝐴𝐵):𝐴𝐶 ↔ (𝑥𝐴𝐵):dom (𝑥𝐴𝐵)⟶𝐶))
72, 6mpbid 235 . . 3 (𝜑 → (𝑥𝐴𝐵):dom (𝑥𝐴𝐵)⟶𝐶)
8 mptelpm.a . . . 4 (𝜑𝐴𝐷)
94, 8eqsstrd 3989 . . 3 (𝜑 → dom (𝑥𝐴𝐵) ⊆ 𝐷)
107, 9jca 515 . 2 (𝜑 → ((𝑥𝐴𝐵):dom (𝑥𝐴𝐵)⟶𝐶 ∧ dom (𝑥𝐴𝐵) ⊆ 𝐷))
11 mptelpm.c . . 3 (𝜑𝐶𝑉)
12 mptelpm.d . . 3 (𝜑𝐷𝑊)
13 elpm2g 8406 . . 3 ((𝐶𝑉𝐷𝑊) → ((𝑥𝐴𝐵) ∈ (𝐶pm 𝐷) ↔ ((𝑥𝐴𝐵):dom (𝑥𝐴𝐵)⟶𝐶 ∧ dom (𝑥𝐴𝐵) ⊆ 𝐷)))
1411, 12, 13syl2anc 587 . 2 (𝜑 → ((𝑥𝐴𝐵) ∈ (𝐶pm 𝐷) ↔ ((𝑥𝐴𝐵):dom (𝑥𝐴𝐵)⟶𝐶 ∧ dom (𝑥𝐴𝐵) ⊆ 𝐷)))
1510, 14mpbird 260 1 (𝜑 → (𝑥𝐴𝐵) ∈ (𝐶pm 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wcel 2115  wss 3918  cmpt 5127  dom cdm 5536  wf 6332  (class class class)co 7138  pm cpm 8390
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-fv 6344  df-ov 7141  df-oprab 7142  df-mpo 7143  df-pm 8392
This theorem is referenced by:  dvnmptconst  42425  dvnmul  42427
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