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Theorem mptssid 41387
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 3509 . . . . . . . 8 (𝑦 = 𝐵𝐵 ∈ V)
21anim2i 616 . . . . . . 7 ((𝑥𝐴𝑦 = 𝐵) → (𝑥𝐴𝐵 ∈ V))
3 rabid 3376 . . . . . . 7 (𝑥 ∈ {𝑥𝐴𝐵 ∈ V} ↔ (𝑥𝐴𝐵 ∈ V))
42, 3sylibr 235 . . . . . 6 ((𝑥𝐴𝑦 = 𝐵) → 𝑥 ∈ {𝑥𝐴𝐵 ∈ V})
5 mptssid.2 . . . . . 6 𝐶 = {𝑥𝐴𝐵 ∈ V}
64, 5eleqtrrdi 2921 . . . . 5 ((𝑥𝐴𝑦 = 𝐵) → 𝑥𝐶)
7 simpr 485 . . . . 5 ((𝑥𝐴𝑦 = 𝐵) → 𝑦 = 𝐵)
86, 7jca 512 . . . 4 ((𝑥𝐴𝑦 = 𝐵) → (𝑥𝐶𝑦 = 𝐵))
9 mptssid.1 . . . . . . . 8 𝑥𝐴
109ssrab2f 41260 . . . . . . 7 {𝑥𝐴𝐵 ∈ V} ⊆ 𝐴
115, 10eqsstri 3998 . . . . . 6 𝐶𝐴
1211sseli 3960 . . . . 5 (𝑥𝐶𝑥𝐴)
1312anim1i 614 . . . 4 ((𝑥𝐶𝑦 = 𝐵) → (𝑥𝐴𝑦 = 𝐵))
148, 13impbii 210 . . 3 ((𝑥𝐴𝑦 = 𝐵) ↔ (𝑥𝐶𝑦 = 𝐵))
1514opabbii 5124 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦 = 𝐵)}
16 df-mpt 5138 . 2 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
17 df-mpt 5138 . 2 (𝑥𝐶𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦 = 𝐵)}
1815, 16, 173eqtr4i 2851 1 (𝑥𝐴𝐵) = (𝑥𝐶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1528  wcel 2105  wnfc 2958  {crab 3139  Vcvv 3492  {copab 5119  cmpt 5137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rab 3144  df-v 3494  df-in 3940  df-ss 3949  df-opab 5120  df-mpt 5138
This theorem is referenced by:  limsupequzmpt2  41875  liminfequzmpt2  41948
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