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Theorem mptelpm 40112
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 6611 . . . 4 (𝜑 → (𝑥𝐴𝐵):𝐴𝐶)
3 eqid 2799 . . . . . . 7 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
43, 1dmmptd 6235 . . . . . 6 (𝜑 → dom (𝑥𝐴𝐵) = 𝐴)
54eqcomd 2805 . . . . 5 (𝜑𝐴 = dom (𝑥𝐴𝐵))
65feq2d 6242 . . . 4 (𝜑 → ((𝑥𝐴𝐵):𝐴𝐶 ↔ (𝑥𝐴𝐵):dom (𝑥𝐴𝐵)⟶𝐶))
72, 6mpbid 224 . . 3 (𝜑 → (𝑥𝐴𝐵):dom (𝑥𝐴𝐵)⟶𝐶)
8 mptelpm.a . . . 4 (𝜑𝐴𝐷)
94, 8eqsstrd 3835 . . 3 (𝜑 → dom (𝑥𝐴𝐵) ⊆ 𝐷)
107, 9jca 508 . 2 (𝜑 → ((𝑥𝐴𝐵):dom (𝑥𝐴𝐵)⟶𝐶 ∧ dom (𝑥𝐴𝐵) ⊆ 𝐷))
11 mptelpm.c . . 3 (𝜑𝐶𝑉)
12 mptelpm.d . . 3 (𝜑𝐷𝑊)
13 elpm2g 8112 . . 3 ((𝐶𝑉𝐷𝑊) → ((𝑥𝐴𝐵) ∈ (𝐶pm 𝐷) ↔ ((𝑥𝐴𝐵):dom (𝑥𝐴𝐵)⟶𝐶 ∧ dom (𝑥𝐴𝐵) ⊆ 𝐷)))
1411, 12, 13syl2anc 580 . 2 (𝜑 → ((𝑥𝐴𝐵) ∈ (𝐶pm 𝐷) ↔ ((𝑥𝐴𝐵):dom (𝑥𝐴𝐵)⟶𝐶 ∧ dom (𝑥𝐴𝐵) ⊆ 𝐷)))
1510, 14mpbird 249 1 (𝜑 → (𝑥𝐴𝐵) ∈ (𝐶pm 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  wcel 2157  wss 3769  cmpt 4922  dom cdm 5312  wf 6097  (class class class)co 6878  pm cpm 8096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-pm 8098
This theorem is referenced by:  dvnmptconst  40900  dvnmul  40902
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