Theorem elsetpreimafvssdm 46564

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 46563	. . 3 ⊢ (𝑆 ∈ 𝑃 → ∃𝑥 ∈ 𝐴 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))
3		cnvimass 6071	. . . . . . . . 9 ⊢ (◡𝐹 “ {(𝐹‘𝑥)}) ⊆ dom 𝐹
4		fndm 6643	. . . . . . . . 9 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
5	3, 4	sseqtrid 4027	. . . . . . . 8 ⊢ (𝐹 Fn 𝐴 → (◡𝐹 “ {(𝐹‘𝑥)}) ⊆ 𝐴)
6	5	adantr 480	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (◡𝐹 “ {(𝐹‘𝑥)}) ⊆ 𝐴)
7		sseq1 4000	. . . . . . 7 ⊢ (𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}) → (𝑆 ⊆ 𝐴 ↔ (◡𝐹 “ {(𝐹‘𝑥)}) ⊆ 𝐴))
8	6, 7	syl5ibrcom 246	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}) → 𝑆 ⊆ 𝐴))
9	8	expcom 413	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (𝐹 Fn 𝐴 → (𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}) → 𝑆 ⊆ 𝐴)))
10	9	com23 86	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}) → (𝐹 Fn 𝐴 → 𝑆 ⊆ 𝐴)))
11	10	rexlimiv 3140	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}) → (𝐹 Fn 𝐴 → 𝑆 ⊆ 𝐴))
12	2, 11	syl 17	. 2 ⊢ (𝑆 ∈ 𝑃 → (𝐹 Fn 𝐴 → 𝑆 ⊆ 𝐴))
13	12	impcom 407	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) → 𝑆 ⊆ 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  {cab 2701  ∃wrex 3062   ⊆ wss 3941  {csn 4621  ^◡ccnv 5666  dom cdm 5667   “ cima 5670   Fn wfn 6529  ‘cfv 6534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-xp 5673  df-cnv 5675  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-fn 6537

This theorem is referenced by:  preimafvsspwdm  46567  uniimaelsetpreimafv  46574