Theorem fvelsetpreimafv 44839

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 44831	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (◡𝐹 “ {(𝐹‘𝑥)}))
2	1	adantrr 714	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))) → 𝑥 ∈ (◡𝐹 “ {(𝐹‘𝑥)}))
3		eleq2 2827	. . . . . . 7 ⊢ (𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}) → (𝑥 ∈ 𝑆 ↔ 𝑥 ∈ (◡𝐹 “ {(𝐹‘𝑥)})))
4	3	ad2antll 726	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))) → (𝑥 ∈ 𝑆 ↔ 𝑥 ∈ (◡𝐹 “ {(𝐹‘𝑥)})))
5	2, 4	mpbird 256	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))) → 𝑥 ∈ 𝑆)
6		simprr 770	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))) → 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))
7	5, 6	jca 512	. . . 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 3199	. 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 44837	. 2 ⊢ (𝑆 ∈ 𝑃 → ∃𝑥 ∈ 𝐴 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))
12	9, 11	impel 506	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) → ∃𝑥 ∈ 𝑆 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  {cab 2715  ∃wrex 3065  {csn 4561  ^◡ccnv 5588   “ cima 5592   Fn wfn 6428  ‘cfv 6433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-fv 6441

This theorem is referenced by:  imaelsetpreimafv  44847