Theorem elsetpreimafvssdm 47846

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 47845	. . 3 ⊢ (𝑆 ∈ 𝑃 → ∃𝑥 ∈ 𝐴 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))
3		cnvimass 6047	. . . . . . . . 9 ⊢ (◡𝐹 “ {(𝐹‘𝑥)}) ⊆ dom 𝐹
4		fndm 6601	. . . . . . . . 9 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
5	3, 4	sseqtrid 3964	. . . . . . . 8 ⊢ (𝐹 Fn 𝐴 → (◡𝐹 “ {(𝐹‘𝑥)}) ⊆ 𝐴)
6	5	adantr 480	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (◡𝐹 “ {(𝐹‘𝑥)}) ⊆ 𝐴)
7		sseq1 3947	. . . . . . 7 ⊢ (𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}) → (𝑆 ⊆ 𝐴 ↔ (◡𝐹 “ {(𝐹‘𝑥)}) ⊆ 𝐴))
8	6, 7	syl5ibrcom 247	. . . . . 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 3131	. . 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 1542   ∈ wcel 2114  {cab 2714  ∃wrex 3061   ⊆ wss 3889  {csn 4567  ^◡ccnv 5630  dom cdm 5631   “ cima 5634   Fn wfn 6493  ‘cfv 6498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-xp 5637  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-fn 6501

This theorem is referenced by:  preimafvsspwdm  47849  uniimaelsetpreimafv  47856