Theorem preimafvsnel 48016

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 489	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ 𝐴)
2		eqidd 2770	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = (𝐹‘𝑋))
3		fniniseg 7056	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝑋 ∈ (◡𝐹 “ {(𝐹‘𝑋)}) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) = (𝐹‘𝑋))))
4	3	adantr 485	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑋 ∈ (◡𝐹 “ {(𝐹‘𝑋)}) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) = (𝐹‘𝑋))))
5	1, 2, 4	mpbir2and 725	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ (◡𝐹 “ {(𝐹‘𝑋)}))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  {csn 4594  ^◡ccnv 5661   “ cima 5665   Fn wfn 6532  ‘cfv 6537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-fv 6545

This theorem is referenced by:  preimafvn0  48017  uniimaprimaeqfv  48019  fvelsetpreimafv  48024  0nelsetpreimafv  48027