Theorem elsetpreimafvssdm 47311

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 47310	. . 3 ⊢ (𝑆 ∈ 𝑃 → ∃𝑥 ∈ 𝐴 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))
3		cnvimass 6102	. . . . . . . . 9 ⊢ (◡𝐹 “ {(𝐹‘𝑥)}) ⊆ dom 𝐹
4		fndm 6672	. . . . . . . . 9 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
5	3, 4	sseqtrid 4048	. . . . . . . 8 ⊢ (𝐹 Fn 𝐴 → (◡𝐹 “ {(𝐹‘𝑥)}) ⊆ 𝐴)
6	5	adantr 480	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (◡𝐹 “ {(𝐹‘𝑥)}) ⊆ 𝐴)
7		sseq1 4021	. . . . . . 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 3146	. . 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 1537   ∈ wcel 2106  {cab 2712  ∃wrex 3068   ⊆ wss 3963  {csn 4631  ^◡ccnv 5688  dom cdm 5689   “ cima 5692   Fn wfn 6558  ‘cfv 6563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-fn 6566

This theorem is referenced by:  preimafvsspwdm  47314  uniimaelsetpreimafv  47321