Theorem uniimaelsetpreimafv 47879

Description: The union of the image of an element of the preimage of a function value is an element of the range of the function. (Contributed by AV, 5-Mar-2024.) (Revised by AV, 22-Mar-2024.)

Hypothesis

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

Assertion

Ref	Expression
uniimaelsetpreimafv	⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) → ∪ (𝐹 “ 𝑆) ∈ ran 𝐹)

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

Allowed substitution hint:   𝑃(𝑧)

Proof of Theorem uniimaelsetpreimafv

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		setpreimafvex.p	. . . . 5 ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}
2	1	0nelsetpreimafv 47873	. . . 4 ⊢ (𝐹 Fn 𝐴 → ∅ ∉ 𝑃)
3		elnelne2 3050	. . . . . 6 ⊢ ((𝑆 ∈ 𝑃 ∧ ∅ ∉ 𝑃) → 𝑆 ≠ ∅)
4		n0 4282	. . . . . 6 ⊢ (𝑆 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝑆)
5	3, 4	sylib 219	. . . . 5 ⊢ ((𝑆 ∈ 𝑃 ∧ ∅ ∉ 𝑃) → ∃𝑦 𝑦 ∈ 𝑆)
6	5	expcom 414	. . . 4 ⊢ (∅ ∉ 𝑃 → (𝑆 ∈ 𝑃 → ∃𝑦 𝑦 ∈ 𝑆))
7	2, 6	syl 17	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝑆 ∈ 𝑃 → ∃𝑦 𝑦 ∈ 𝑆))
8	7	imp 407	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) → ∃𝑦 𝑦 ∈ 𝑆)
9	1	imaelsetpreimafv 47878	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑦 ∈ 𝑆) → (𝐹 “ 𝑆) = {(𝐹‘𝑦)})
10	9	3expa 1124	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) ∧ 𝑦 ∈ 𝑆) → (𝐹 “ 𝑆) = {(𝐹‘𝑦)})
11	10	unieqd 4852	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) ∧ 𝑦 ∈ 𝑆) → ∪ (𝐹 “ 𝑆) = ∪ {(𝐹‘𝑦)})
12		fvex 6841	. . . . 5 ⊢ (𝐹‘𝑦) ∈ V
13	12	unisn 4858	. . . 4 ⊢ ∪ {(𝐹‘𝑦)} = (𝐹‘𝑦)
14	11, 13	eqtrdi 2790	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) ∧ 𝑦 ∈ 𝑆) → ∪ (𝐹 “ 𝑆) = (𝐹‘𝑦))
15		dffn3 6668	. . . . . 6 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)
16	15	biimpi 217	. . . . 5 ⊢ (𝐹 Fn 𝐴 → 𝐹:𝐴⟶ran 𝐹)
17	16	ad2antrr 732	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) ∧ 𝑦 ∈ 𝑆) → 𝐹:𝐴⟶ran 𝐹)
18	1	elsetpreimafvssdm 47869	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) → 𝑆 ⊆ 𝐴)
19	18	sselda 3915	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ 𝐴)
20	17, 19	ffvelcdmd 7027	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) ∧ 𝑦 ∈ 𝑆) → (𝐹‘𝑦) ∈ ran 𝐹)
21	14, 20	eqeltrd 2839	. 2 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) ∧ 𝑦 ∈ 𝑆) → ∪ (𝐹 “ 𝑆) ∈ ran 𝐹)
22	8, 21	exlimddv 1942	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) → ∪ (𝐹 “ 𝑆) ∈ ran 𝐹)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547  ∃wex 1786   ∈ wcel 2119  {cab 2717   ≠ wne 2934   ∉ wnel 3038  ∃wrex 3063  ∅c0 4262  {csn 4556  ∪ cuni 4839  ^◡ccnv 5618  ran crn 5620   “ cima 5622   Fn wfn 6481  ⟶wf 6482  ‘cfv 6486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-12 2189  ax-ext 2711  ax-sep 5219  ax-nul 5229  ax-pr 5363

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-br 5074  df-opab 5136  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-fv 6494

This theorem is referenced by:  imasetpreimafvbijlemf  47884  fundcmpsurbijinjpreimafv  47890