Theorem mptmex 5939

Description: If a function given by maps-to notation is inhabited, then the class it is defined on is inhabited. (Contributed by Jim Kingdon, 22-Jul-2026.)

Assertion

Ref	Expression
mptmex	⊢ (𝐶 ∈ (𝑥 ∈ 𝐴 ↦ 𝐵) → ∃𝑦 𝑦 ∈ 𝐴)

Distinct variable group:   𝑥,𝐴,𝑦

Allowed substitution hints:   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem mptmex

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex2 2838	. . . . 5 ⊢ (𝐶 ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)} → ∃𝑤 𝑤 ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)})
2		opabm 4421	. . . . 5 ⊢ (∃𝑤 𝑤 ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)} ↔ ∃𝑥∃𝑧(𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵))
3	1, 2	sylib 122	. . . 4 ⊢ (𝐶 ∈ {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)} → ∃𝑥∃𝑧(𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵))
4		df-mpt 4192	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)}
5	3, 4	eleq2s 2333	. . 3 ⊢ (𝐶 ∈ (𝑥 ∈ 𝐴 ↦ 𝐵) → ∃𝑥∃𝑧(𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵))
6		simpl 109	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) → 𝑥 ∈ 𝐴)
7	6	exlimiv 1651	. . . 4 ⊢ (∃𝑧(𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) → 𝑥 ∈ 𝐴)
8	7	eximi 1653	. . 3 ⊢ (∃𝑥∃𝑧(𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) → ∃𝑥 𝑥 ∈ 𝐴)
9	5, 8	syl 14	. 2 ⊢ (𝐶 ∈ (𝑥 ∈ 𝐴 ↦ 𝐵) → ∃𝑥 𝑥 ∈ 𝐴)
10		eleq1w 2299	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
11	10	cbvexv 1974	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑦 𝑦 ∈ 𝐴)
12	9, 11	sylib 122	1 ⊢ (𝐶 ∈ (𝑥 ∈ 𝐴 ↦ 𝐵) → ∃𝑦 𝑦 ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1402  ∃wex 1545   ∈ wcel 2209  {copab 4189   ↦ cmpt 4190

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-opab 4191  df-mpt 4192

This theorem is referenced by:  asclfval  15004