Theorem elsetpreimafvssdm 48273

Description: An element of the class 𝑃 of all preimages of function values is a subset of the domain of the function. (Contributed by AV, 8-Mar-2024.)

Hypothesis

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

Assertion

Ref	Expression
elsetpreimafvssdm	⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) → 𝑆 ⊆ 𝐴)

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

Allowed substitution hints:   𝑃(𝑥, 𝑧)

Proof of Theorem elsetpreimafvssdm

Step	Hyp	Ref	Expression
1		setpreimafvex.p	. . . 4 ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}
2	1	elsetpreimafv 48272	. . 3 ⊢ (𝑆 ∈ 𝑃 → ∃𝑥 ∈ 𝐴 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))
3		cnvimass 6082	. . . . . . . . 9 ⊢ (◡𝐹 “ {(𝐹‘𝑥)}) ⊆ dom 𝐹
4		fndm 6639	. . . . . . . . 9 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
5	3, 4	sseqtrid 3976	. . . . . . . 8 ⊢ (𝐹 Fn 𝐴 → (◡𝐹 “ {(𝐹‘𝑥)}) ⊆ 𝐴)
6	5	adantr 486	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (◡𝐹 “ {(𝐹‘𝑥)}) ⊆ 𝐴)
7		sseq1 3959	. . . . . . 7 ⊢ (𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}) → (𝑆 ⊆ 𝐴 ↔ (◡𝐹 “ {(𝐹‘𝑥)}) ⊆ 𝐴))
8	6, 7	syl5ibrcom 250	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}) → 𝑆 ⊆ 𝐴))
9	8	expcom 419	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (𝐹 Fn 𝐴 → (𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}) → 𝑆 ⊆ 𝐴)))
10	9	com23 87	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}) → (𝐹 Fn 𝐴 → 𝑆 ⊆ 𝐴)))
11	10	rexlimiv 3158	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}) → (𝐹 Fn 𝐴 → 𝑆 ⊆ 𝐴))
12	2, 11	syl 18	. 2 ⊢ (𝑆 ∈ 𝑃 → (𝐹 Fn 𝐴 → 𝑆 ⊆ 𝐴))
13	12	impcom 413	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) → 𝑆 ⊆ 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2740  ∃wrex 3088   ⊆ wss 3902  {csn 4587  ^◡ccnv 5658  dom cdm 5659   “ cima 5662   Fn wfn 6532  ‘cfv 6537

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-xp 5665  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-fn 6540

This theorem is used by:  preimafvsspwdm  48276  uniimaelsetpreimafv  48283