Theorem fvelsetpreimafv 47862

Description: There is an element in a preimage 𝑆 of function values so that 𝑆 is the preimage of the function value at this element. (Contributed by AV, 8-Mar-2024.)

Hypothesis

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

Assertion

Ref	Expression
fvelsetpreimafv	⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) → ∃𝑥 ∈ 𝑆 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))

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

Allowed substitution hints:   𝑃(𝑥,𝑧)

Proof of Theorem fvelsetpreimafv

Step	Hyp	Ref	Expression
1		preimafvsnel 47854	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (◡𝐹 “ {(𝐹‘𝑥)}))
2	1	adantrr 723	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))) → 𝑥 ∈ (◡𝐹 “ {(𝐹‘𝑥)}))
3		eleq2 2828	. . . . . . 7 ⊢ (𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}) → (𝑥 ∈ 𝑆 ↔ 𝑥 ∈ (◡𝐹 “ {(𝐹‘𝑥)})))
4	3	ad2antll 735	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))) → (𝑥 ∈ 𝑆 ↔ 𝑥 ∈ (◡𝐹 “ {(𝐹‘𝑥)})))
5	2, 4	mpbird 258	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))) → 𝑥 ∈ 𝑆)
6		simprr 778	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))) → 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))
7	5, 6	jca 516	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))) → (𝑥 ∈ 𝑆 ∧ 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)})))
8	7	ex 413	. . 3 ⊢ (𝐹 Fn 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)})) → (𝑥 ∈ 𝑆 ∧ 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))))
9	8	reximdv2 3149	. 2 ⊢ (𝐹 Fn 𝐴 → (∃𝑥 ∈ 𝐴 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}) → ∃𝑥 ∈ 𝑆 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)})))
10		setpreimafvex.p	. . 3 ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}
11	10	elsetpreimafv 47860	. 2 ⊢ (𝑆 ∈ 𝑃 → ∃𝑥 ∈ 𝐴 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))
12	9, 11	impel 510	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) → ∃𝑥 ∈ 𝑆 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {cab 2717  ∃wrex 3063  {csn 4555  ^◡ccnv 5617   “ cima 5621   Fn wfn 6480  ‘cfv 6485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-fv 6493

This theorem is referenced by:  imaelsetpreimafv  47870