Theorem elfvm 5726

Description: If a function value has a member, the function is inhabited. (Contributed by Jim Kingdon, 14-Jun-2025.)

Assertion

Ref	Expression
elfvm	⊢ (𝐴 ∈ (𝐹‘𝐵) → ∃𝑗 𝑗 ∈ 𝐹)

Distinct variable group:   𝑗,𝐹

Allowed substitution hints:   𝐴(𝑗)   𝐵(𝑗)

Proof of Theorem elfvm

Dummy variables 𝑘 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eliotaeu 5364	. . . 4 ⊢ (𝐴 ∈ (℩𝑥𝐵𝐹𝑥) → ∃!𝑥 𝐵𝐹𝑥)
2		df-fv 5383	. . . 4 ⊢ (𝐹‘𝐵) = (℩𝑥𝐵𝐹𝑥)
3	1, 2	eleq2s 2333	. . 3 ⊢ (𝐴 ∈ (𝐹‘𝐵) → ∃!𝑥 𝐵𝐹𝑥)
4		euex 2116	. . 3 ⊢ (∃!𝑥 𝐵𝐹𝑥 → ∃𝑥 𝐵𝐹𝑥)
5		brm 4179	. . . 4 ⊢ (𝐵𝐹𝑥 → ∃𝑘 𝑘 ∈ 𝐹)
6	5	exlimiv 1651	. . 3 ⊢ (∃𝑥 𝐵𝐹𝑥 → ∃𝑘 𝑘 ∈ 𝐹)
7	3, 4, 6	3syl 17	. 2 ⊢ (𝐴 ∈ (𝐹‘𝐵) → ∃𝑘 𝑘 ∈ 𝐹)
8		eleq1w 2299	. . 3 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝐹 ↔ 𝑗 ∈ 𝐹))
9	8	cbvexv 1974	. 2 ⊢ (∃𝑘 𝑘 ∈ 𝐹 ↔ ∃𝑗 𝑗 ∈ 𝐹)
10	7, 9	sylib 122	1 ⊢ (𝐴 ∈ (𝐹‘𝐵) → ∃𝑗 𝑗 ∈ 𝐹)

Syntax hints:   → wi 4  ∃wex 1545  ∃!weu 2086   ∈ wcel 2209   class class class wbr 4128  ℩cio 5333  ‘cfv 5375

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-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-sn 3714  df-uni 3934  df-br 4129  df-iota 5335  df-fv 5383

This theorem is referenced by:  basm  13397  slotm  13398