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Theorem mptssid 42045
 Description: The mapping operation expressed with its actual domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
mptssid.1 𝑥𝐴
mptssid.2 𝐶 = {𝑥𝐴𝐵 ∈ V}
Assertion
Ref Expression
mptssid (𝑥𝐴𝐵) = (𝑥𝐶𝐵)

Proof of Theorem mptssid
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eqvisset 3459 . . . . . . . 8 (𝑦 = 𝐵𝐵 ∈ V)
21anim2i 619 . . . . . . 7 ((𝑥𝐴𝑦 = 𝐵) → (𝑥𝐴𝐵 ∈ V))
3 rabid 3332 . . . . . . 7 (𝑥 ∈ {𝑥𝐴𝐵 ∈ V} ↔ (𝑥𝐴𝐵 ∈ V))
42, 3sylibr 237 . . . . . 6 ((𝑥𝐴𝑦 = 𝐵) → 𝑥 ∈ {𝑥𝐴𝐵 ∈ V})
5 mptssid.2 . . . . . 6 𝐶 = {𝑥𝐴𝐵 ∈ V}
64, 5eleqtrrdi 2901 . . . . 5 ((𝑥𝐴𝑦 = 𝐵) → 𝑥𝐶)
7 simpr 488 . . . . 5 ((𝑥𝐴𝑦 = 𝐵) → 𝑦 = 𝐵)
86, 7jca 515 . . . 4 ((𝑥𝐴𝑦 = 𝐵) → (𝑥𝐶𝑦 = 𝐵))
9 mptssid.1 . . . . . . . 8 𝑥𝐴
109ssrab2f 41923 . . . . . . 7 {𝑥𝐴𝐵 ∈ V} ⊆ 𝐴
115, 10eqsstri 3951 . . . . . 6 𝐶𝐴
1211sseli 3913 . . . . 5 (𝑥𝐶𝑥𝐴)
1312anim1i 617 . . . 4 ((𝑥𝐶𝑦 = 𝐵) → (𝑥𝐴𝑦 = 𝐵))
148, 13impbii 212 . . 3 ((𝑥𝐴𝑦 = 𝐵) ↔ (𝑥𝐶𝑦 = 𝐵))
1514opabbii 5101 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦 = 𝐵)}
16 df-mpt 5115 . 2 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
17 df-mpt 5115 . 2 (𝑥𝐶𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦 = 𝐵)}
1815, 16, 173eqtr4i 2831 1 (𝑥𝐴𝐵) = (𝑥𝐶𝐵)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Ⅎwnfc 2936  {crab 3110  Vcvv 3442  {copab 5096   ↦ cmpt 5114 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rab 3115  df-v 3444  df-in 3890  df-ss 3900  df-opab 5097  df-mpt 5115 This theorem is referenced by:  limsupequzmpt2  42528  liminfequzmpt2  42601
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