Theorem elsetpreimafvssdm 48058

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 48057	. . 3 ⊢ (𝑆 ∈ 𝑃 → ∃𝑥 ∈ 𝐴 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))
3		cnvimass 6085	. . . . . . . . 9 ⊢ (◡𝐹 “ {(𝐹‘𝑥)}) ⊆ dom 𝐹
4		fndm 6639	. . . . . . . . 9 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
5	3, 4	sseqtrid 3987	. . . . . . . 8 ⊢ (𝐹 Fn 𝐴 → (◡𝐹 “ {(𝐹‘𝑥)}) ⊆ 𝐴)
6	5	adantr 485	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (◡𝐹 “ {(𝐹‘𝑥)}) ⊆ 𝐴)
7		sseq1 3970	. . . . . . 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 418	. . . . 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 3165	. . 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 412	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) → 𝑆 ⊆ 𝐴)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  {cab 2747  ∃wrex 3095   ⊆ wss 3913  {csn 4594  ^◡ccnv 5661  dom cdm 5662   “ cima 5665   Fn wfn 6532  ‘cfv 6537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-xp 5668  df-cnv 5670  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-fn 6540

This theorem is referenced by:  preimafvsspwdm  48061  uniimaelsetpreimafv  48068