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Theorem mptssid 42457
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 3425 . . . . . . . 8 (𝑦 = 𝐵𝐵 ∈ V)
21anim2i 620 . . . . . . 7 ((𝑥𝐴𝑦 = 𝐵) → (𝑥𝐴𝐵 ∈ V))
3 rabid 3290 . . . . . . 7 (𝑥 ∈ {𝑥𝐴𝐵 ∈ V} ↔ (𝑥𝐴𝐵 ∈ V))
42, 3sylibr 237 . . . . . 6 ((𝑥𝐴𝑦 = 𝐵) → 𝑥 ∈ {𝑥𝐴𝐵 ∈ V})
5 mptssid.2 . . . . . 6 𝐶 = {𝑥𝐴𝐵 ∈ V}
64, 5eleqtrrdi 2849 . . . . 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 42339 . . . . . . 7 {𝑥𝐴𝐵 ∈ V} ⊆ 𝐴
115, 10eqsstri 3935 . . . . . 6 𝐶𝐴
1211sseli 3896 . . . . 5 (𝑥𝐶𝑥𝐴)
1312anim1i 618 . . . 4 ((𝑥𝐶𝑦 = 𝐵) → (𝑥𝐴𝑦 = 𝐵))
148, 13impbii 212 . . 3 ((𝑥𝐴𝑦 = 𝐵) ↔ (𝑥𝐶𝑦 = 𝐵))
1514opabbii 5120 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦 = 𝐵)}
16 df-mpt 5136 . 2 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
17 df-mpt 5136 . 2 (𝑥𝐶𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦 = 𝐵)}
1815, 16, 173eqtr4i 2775 1 (𝑥𝐴𝐵) = (𝑥𝐶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1543  wcel 2110  wnfc 2884  {crab 3065  Vcvv 3408  {copab 5115  cmpt 5135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rab 3070  df-v 3410  df-in 3873  df-ss 3883  df-opab 5116  df-mpt 5136
This theorem is referenced by:  limsupequzmpt2  42934  liminfequzmpt2  43007
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