Theorem elfvdmOLD 6529

Description: Obsolete proof of elfvdm 6528 as of 22-Oct-2022. (Contributed by NM, 12-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
elfvdmOLD	⊢ (𝐴 ∈ (𝐹‘𝐵) → 𝐵 ∈ dom 𝐹)

Proof of Theorem elfvdmOLD

Step	Hyp	Ref	Expression
1		ne0i 4180	. 2 ⊢ (𝐴 ∈ (𝐹‘𝐵) → (𝐹‘𝐵) ≠ ∅)
2		ndmfv 6526	. . 3 ⊢ (¬ 𝐵 ∈ dom 𝐹 → (𝐹‘𝐵) = ∅)
3	2	necon1ai 2987	. 2 ⊢ ((𝐹‘𝐵) ≠ ∅ → 𝐵 ∈ dom 𝐹)
4	1, 3	syl 17	1 ⊢ (𝐴 ∈ (𝐹‘𝐵) → 𝐵 ∈ dom 𝐹)

Syntax hints:   → wi 4   ∈ wcel 2051   ≠ wne 2960  ∅c0 4172  dom cdm 5403  ‘cfv 6185

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2743  ax-nul 5063  ax-pow 5115

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-dm 5413  df-iota 6149  df-fv 6193

This theorem is referenced by: (None)