Theorem fvelsetpreimafv 47575

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 47567	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (◡𝐹 “ {(𝐹‘𝑥)}))
2	1	adantrr 717	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))) → 𝑥 ∈ (◡𝐹 “ {(𝐹‘𝑥)}))
3		eleq2 2823	. . . . . . 7 ⊢ (𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}) → (𝑥 ∈ 𝑆 ↔ 𝑥 ∈ (◡𝐹 “ {(𝐹‘𝑥)})))
4	3	ad2antll 729	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))) → (𝑥 ∈ 𝑆 ↔ 𝑥 ∈ (◡𝐹 “ {(𝐹‘𝑥)})))
5	2, 4	mpbird 257	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))) → 𝑥 ∈ 𝑆)
6		simprr 772	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))) → 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))
7	5, 6	jca 511	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))) → (𝑥 ∈ 𝑆 ∧ 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)})))
8	7	ex 412	. . 3 ⊢ (𝐹 Fn 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)})) → (𝑥 ∈ 𝑆 ∧ 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))))
9	8	reximdv2 3144	. 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 47573	. 2 ⊢ (𝑆 ∈ 𝑃 → ∃𝑥 ∈ 𝐴 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))
12	9, 11	impel 505	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) → ∃𝑥 ∈ 𝑆 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {cab 2712  ∃wrex 3058  {csn 4578  ^◡ccnv 5621   “ cima 5625   Fn wfn 6485  ‘cfv 6490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-fv 6498

This theorem is referenced by:  imaelsetpreimafv  47583