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Theorem mptelpm 39875
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)
2 eqid 2761 . . . . 5 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
31, 2fmptd 6550 . . . 4 (𝜑 → (𝑥𝐴𝐵):𝐴𝐶)
42, 1dmmptd 6186 . . . . . 6 (𝜑 → dom (𝑥𝐴𝐵) = 𝐴)
54eqcomd 2767 . . . . 5 (𝜑𝐴 = dom (𝑥𝐴𝐵))
65feq2d 6193 . . . 4 (𝜑 → ((𝑥𝐴𝐵):𝐴𝐶 ↔ (𝑥𝐴𝐵):dom (𝑥𝐴𝐵)⟶𝐶))
73, 6mpbid 222 . . 3 (𝜑 → (𝑥𝐴𝐵):dom (𝑥𝐴𝐵)⟶𝐶)
8 mptelpm.a . . . 4 (𝜑𝐴𝐷)
94, 8eqsstrd 3781 . . 3 (𝜑 → dom (𝑥𝐴𝐵) ⊆ 𝐷)
107, 9jca 555 . 2 (𝜑 → ((𝑥𝐴𝐵):dom (𝑥𝐴𝐵)⟶𝐶 ∧ dom (𝑥𝐴𝐵) ⊆ 𝐷))
11 mptelpm.c . . 3 (𝜑𝐶𝑉)
12 mptelpm.d . . 3 (𝜑𝐷𝑊)
13 elpm2g 8043 . . 3 ((𝐶𝑉𝐷𝑊) → ((𝑥𝐴𝐵) ∈ (𝐶pm 𝐷) ↔ ((𝑥𝐴𝐵):dom (𝑥𝐴𝐵)⟶𝐶 ∧ dom (𝑥𝐴𝐵) ⊆ 𝐷)))
1411, 12, 13syl2anc 696 . 2 (𝜑 → ((𝑥𝐴𝐵) ∈ (𝐶pm 𝐷) ↔ ((𝑥𝐴𝐵):dom (𝑥𝐴𝐵)⟶𝐶 ∧ dom (𝑥𝐴𝐵) ⊆ 𝐷)))
1510, 14mpbird 247 1 (𝜑 → (𝑥𝐴𝐵) ∈ (𝐶pm 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wcel 2140  wss 3716  cmpt 4882  dom cdm 5267  wf 6046  (class class class)co 6815  pm cpm 8027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-fv 6058  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-pm 8029
This theorem is referenced by:  dvnmptconst  40678  dvnmul  40680
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