Theorem elsetpreimafvrab 48001

Description: An element of the preimage of a function value expressed as a restricted class abstraction. (Contributed by AV, 9-Mar-2024.)

Hypothesis

Ref	Expression
setpreimafvex.p	⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}

Assertion

Ref	Expression
elsetpreimafvrab	⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) → 𝑆 = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐹‘𝑋)})

Distinct variable groups:   𝑥,𝐴,𝑧   𝑥,𝐹,𝑧   𝑥,𝑆,𝑧   𝑥,𝑋

Allowed substitution hints:   𝑃(𝑥,𝑧)   𝑋(𝑧)

Proof of Theorem elsetpreimafvrab

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		setpreimafvex.p	. . . 4 ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}
2	1	elsetpreimafvbi 47998	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) → (𝑦 ∈ 𝑆 ↔ (𝑦 ∈ 𝐴 ∧ (𝐹‘𝑦) = (𝐹‘𝑋))))
3		fveqeq2 6877	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) = (𝐹‘𝑋) ↔ (𝐹‘𝑦) = (𝐹‘𝑋)))
4	3	elrab 3651	. . 3 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐹‘𝑋)} ↔ (𝑦 ∈ 𝐴 ∧ (𝐹‘𝑦) = (𝐹‘𝑋)))
5	2, 4	bitr4di 291	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) → (𝑦 ∈ 𝑆 ↔ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐹‘𝑋)}))
6	5	eqrdv 2761	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) → 𝑆 = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐹‘𝑋)})

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1099   = wceq 1561   ∈ wcel 2143  {cab 2741  ∃wrex 3087  {crab 3415  {csn 4583  ^◡ccnv 5647   “ cima 5651   Fn wfn 6517  ‘cfv 6522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-nul 5257  ax-pr 5391

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5102  df-opab 5164  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-iota 6478  df-fun 6524  df-fn 6525  df-fv 6530

This theorem is referenced by:  elsetpreimafveq  48004