Theorem preimafvsnel 47982

Description: The preimage of a function value at 𝑋 contains 𝑋. (Contributed by AV, 7-Mar-2024.)

Assertion

Ref	Expression
preimafvsnel	⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ (◡𝐹 “ {(𝐹‘𝑋)}))

Proof of Theorem preimafvsnel

Step	Hyp	Ref	Expression
1		simpr 488	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ 𝐴)
2		eqidd 2763	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = (𝐹‘𝑋))
3		fniniseg 7041	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝑋 ∈ (◡𝐹 “ {(𝐹‘𝑋)}) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) = (𝐹‘𝑋))))
4	3	adantr 484	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑋 ∈ (◡𝐹 “ {(𝐹‘𝑋)}) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) = (𝐹‘𝑋))))
5	1, 2, 4	mpbir2and 723	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ (◡𝐹 “ {(𝐹‘𝑋)}))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1560   ∈ wcel 2142  {csn 4582  ^◡ccnv 5646   “ cima 5650   Fn wfn 6516  ‘cfv 6521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-fv 6529

This theorem is referenced by:  preimafvn0  47983  uniimaprimaeqfv  47985  fvelsetpreimafv  47990  0nelsetpreimafv  47993