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Theorem mptssid 39967
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 simpl 474 . . . . . . . . . 10 ((𝑥𝐴𝑦 = 𝐵) → 𝑥𝐴)
2 eqvisset 3351 . . . . . . . . . . 11 (𝑦 = 𝐵𝐵 ∈ V)
32adantl 473 . . . . . . . . . 10 ((𝑥𝐴𝑦 = 𝐵) → 𝐵 ∈ V)
41, 3jca 555 . . . . . . . . 9 ((𝑥𝐴𝑦 = 𝐵) → (𝑥𝐴𝐵 ∈ V))
5 rabid 3254 . . . . . . . . 9 (𝑥 ∈ {𝑥𝐴𝐵 ∈ V} ↔ (𝑥𝐴𝐵 ∈ V))
64, 5sylibr 224 . . . . . . . 8 ((𝑥𝐴𝑦 = 𝐵) → 𝑥 ∈ {𝑥𝐴𝐵 ∈ V})
7 mptssid.2 . . . . . . . 8 𝐶 = {𝑥𝐴𝐵 ∈ V}
86, 7syl6eleqr 2850 . . . . . . 7 ((𝑥𝐴𝑦 = 𝐵) → 𝑥𝐶)
9 simpr 479 . . . . . . 7 ((𝑥𝐴𝑦 = 𝐵) → 𝑦 = 𝐵)
108, 9jca 555 . . . . . 6 ((𝑥𝐴𝑦 = 𝐵) → (𝑥𝐶𝑦 = 𝐵))
11 mptssid.1 . . . . . . . . . . 11 𝑥𝐴
1211ssrab2f 39817 . . . . . . . . . 10 {𝑥𝐴𝐵 ∈ V} ⊆ 𝐴
137, 12eqsstri 3776 . . . . . . . . 9 𝐶𝐴
14 id 22 . . . . . . . . 9 (𝑥𝐶𝑥𝐶)
1513, 14sseldi 3742 . . . . . . . 8 (𝑥𝐶𝑥𝐴)
1615adantr 472 . . . . . . 7 ((𝑥𝐶𝑦 = 𝐵) → 𝑥𝐴)
17 simpr 479 . . . . . . 7 ((𝑥𝐶𝑦 = 𝐵) → 𝑦 = 𝐵)
1816, 17jca 555 . . . . . 6 ((𝑥𝐶𝑦 = 𝐵) → (𝑥𝐴𝑦 = 𝐵))
1910, 18impbii 199 . . . . 5 ((𝑥𝐴𝑦 = 𝐵) ↔ (𝑥𝐶𝑦 = 𝐵))
2019ax-gen 1871 . . . 4 𝑦((𝑥𝐴𝑦 = 𝐵) ↔ (𝑥𝐶𝑦 = 𝐵))
2120ax-gen 1871 . . 3 𝑥𝑦((𝑥𝐴𝑦 = 𝐵) ↔ (𝑥𝐶𝑦 = 𝐵))
22 eqopab2b 5155 . . 3 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦 = 𝐵)} ↔ ∀𝑥𝑦((𝑥𝐴𝑦 = 𝐵) ↔ (𝑥𝐶𝑦 = 𝐵)))
2321, 22mpbir 221 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦 = 𝐵)}
24 df-mpt 4882 . 2 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
25 df-mpt 4882 . 2 (𝑥𝐶𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦 = 𝐵)}
2623, 24, 253eqtr4i 2792 1 (𝑥𝐴𝐵) = (𝑥𝐶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  wal 1630   = wceq 1632  wcel 2139  wnfc 2889  {crab 3054  Vcvv 3340  {copab 4864  cmpt 4881
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 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
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 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-opab 4865  df-mpt 4882
This theorem is referenced by:  limsupequzmpt2  40471  liminfequzmpt2  40544
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