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Theorem mptssid 43944
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 3492 . . . . . . . 8 (𝑦 = 𝐵𝐵 ∈ V)
21anim2i 618 . . . . . . 7 ((𝑥𝐴𝑦 = 𝐵) → (𝑥𝐴𝐵 ∈ V))
3 rabid 3453 . . . . . . 7 (𝑥 ∈ {𝑥𝐴𝐵 ∈ V} ↔ (𝑥𝐴𝐵 ∈ V))
42, 3sylibr 233 . . . . . 6 ((𝑥𝐴𝑦 = 𝐵) → 𝑥 ∈ {𝑥𝐴𝐵 ∈ V})
5 mptssid.2 . . . . . 6 𝐶 = {𝑥𝐴𝐵 ∈ V}
64, 5eleqtrrdi 2845 . . . . 5 ((𝑥𝐴𝑦 = 𝐵) → 𝑥𝐶)
7 simpr 486 . . . . 5 ((𝑥𝐴𝑦 = 𝐵) → 𝑦 = 𝐵)
86, 7jca 513 . . . 4 ((𝑥𝐴𝑦 = 𝐵) → (𝑥𝐶𝑦 = 𝐵))
9 mptssid.1 . . . . . . . 8 𝑥𝐴
109ssrab2f 43806 . . . . . . 7 {𝑥𝐴𝐵 ∈ V} ⊆ 𝐴
115, 10eqsstri 4017 . . . . . 6 𝐶𝐴
1211sseli 3979 . . . . 5 (𝑥𝐶𝑥𝐴)
1312anim1i 616 . . . 4 ((𝑥𝐶𝑦 = 𝐵) → (𝑥𝐴𝑦 = 𝐵))
148, 13impbii 208 . . 3 ((𝑥𝐴𝑦 = 𝐵) ↔ (𝑥𝐶𝑦 = 𝐵))
1514opabbii 5216 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦 = 𝐵)}
16 df-mpt 5233 . 2 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
17 df-mpt 5233 . 2 (𝑥𝐶𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦 = 𝐵)}
1815, 16, 173eqtr4i 2771 1 (𝑥𝐴𝐵) = (𝑥𝐶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wa 397   = wceq 1542  wcel 2107  wnfc 2884  {crab 3433  Vcvv 3475  {copab 5211  cmpt 5232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rab 3434  df-v 3477  df-in 3956  df-ss 3966  df-opab 5212  df-mpt 5233
This theorem is referenced by:  limsupequzmpt2  44434  liminfequzmpt2  44507
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