Theorem elfvm 5619

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 5266	. . . 4 ⊢ (𝐴 ∈ (℩𝑥𝐵𝐹𝑥) → ∃!𝑥 𝐵𝐹𝑥)
2		df-fv 5285	. . . 4 ⊢ (𝐹‘𝐵) = (℩𝑥𝐵𝐹𝑥)
3	1, 2	eleq2s 2301	. . 3 ⊢ (𝐴 ∈ (𝐹‘𝐵) → ∃!𝑥 𝐵𝐹𝑥)
4		euex 2085	. . 3 ⊢ (∃!𝑥 𝐵𝐹𝑥 → ∃𝑥 𝐵𝐹𝑥)
5		brm 4099	. . . 4 ⊢ (𝐵𝐹𝑥 → ∃𝑘 𝑘 ∈ 𝐹)
6	5	exlimiv 1622	. . 3 ⊢ (∃𝑥 𝐵𝐹𝑥 → ∃𝑘 𝑘 ∈ 𝐹)
7	3, 4, 6	3syl 17	. 2 ⊢ (𝐴 ∈ (𝐹‘𝐵) → ∃𝑘 𝑘 ∈ 𝐹)
8		eleq1w 2267	. . 3 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝐹 ↔ 𝑗 ∈ 𝐹))
9	8	cbvexv 1943	. 2 ⊢ (∃𝑘 𝑘 ∈ 𝐹 ↔ ∃𝑗 𝑗 ∈ 𝐹)
10	7, 9	sylib 122	1 ⊢ (𝐴 ∈ (𝐹‘𝐵) → ∃𝑗 𝑗 ∈ 𝐹)

Syntax hints:   → wi 4  ∃wex 1516  ∃!weu 2055   ∈ wcel 2177   class class class wbr 4048  ℩cio 5236  ‘cfv 5277

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-rex 2491  df-v 2775  df-sn 3641  df-uni 3854  df-br 4049  df-iota 5238  df-fv 5285

This theorem is referenced by:  basm  12943