Theorem elfvm 5656

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 5303	. . . 4 ⊢ (𝐴 ∈ (℩𝑥𝐵𝐹𝑥) → ∃!𝑥 𝐵𝐹𝑥)
2		df-fv 5322	. . . 4 ⊢ (𝐹‘𝐵) = (℩𝑥𝐵𝐹𝑥)
3	1, 2	eleq2s 2324	. . 3 ⊢ (𝐴 ∈ (𝐹‘𝐵) → ∃!𝑥 𝐵𝐹𝑥)
4		euex 2107	. . 3 ⊢ (∃!𝑥 𝐵𝐹𝑥 → ∃𝑥 𝐵𝐹𝑥)
5		brm 4133	. . . 4 ⊢ (𝐵𝐹𝑥 → ∃𝑘 𝑘 ∈ 𝐹)
6	5	exlimiv 1644	. . 3 ⊢ (∃𝑥 𝐵𝐹𝑥 → ∃𝑘 𝑘 ∈ 𝐹)
7	3, 4, 6	3syl 17	. 2 ⊢ (𝐴 ∈ (𝐹‘𝐵) → ∃𝑘 𝑘 ∈ 𝐹)
8		eleq1w 2290	. . 3 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝐹 ↔ 𝑗 ∈ 𝐹))
9	8	cbvexv 1965	. 2 ⊢ (∃𝑘 𝑘 ∈ 𝐹 ↔ ∃𝑗 𝑗 ∈ 𝐹)
10	7, 9	sylib 122	1 ⊢ (𝐴 ∈ (𝐹‘𝐵) → ∃𝑗 𝑗 ∈ 𝐹)

Syntax hints:   → wi 4  ∃wex 1538  ∃!weu 2077   ∈ wcel 2200   class class class wbr 4082  ℩cio 5272  ‘cfv 5314

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-sn 3672  df-uni 3888  df-br 4083  df-iota 5274  df-fv 5322

This theorem is referenced by:  basm  13080