Theorem uniimaelsetpreimafv 47390

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 47384	. . . 4 ⊢ (𝐹 Fn 𝐴 → ∅ ∉ 𝑃)
3		elnelne2 3041	. . . . . 6 ⊢ ((𝑆 ∈ 𝑃 ∧ ∅ ∉ 𝑃) → 𝑆 ≠ ∅)
4		n0 4312	. . . . . 6 ⊢ (𝑆 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝑆)
5	3, 4	sylib 218	. . . . 5 ⊢ ((𝑆 ∈ 𝑃 ∧ ∅ ∉ 𝑃) → ∃𝑦 𝑦 ∈ 𝑆)
6	5	expcom 413	. . . 4 ⊢ (∅ ∉ 𝑃 → (𝑆 ∈ 𝑃 → ∃𝑦 𝑦 ∈ 𝑆))
7	2, 6	syl 17	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝑆 ∈ 𝑃 → ∃𝑦 𝑦 ∈ 𝑆))
8	7	imp 406	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) → ∃𝑦 𝑦 ∈ 𝑆)
9	1	imaelsetpreimafv 47389	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑦 ∈ 𝑆) → (𝐹 “ 𝑆) = {(𝐹‘𝑦)})
10	9	3expa 1118	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) ∧ 𝑦 ∈ 𝑆) → (𝐹 “ 𝑆) = {(𝐹‘𝑦)})
11	10	unieqd 4880	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) ∧ 𝑦 ∈ 𝑆) → ∪ (𝐹 “ 𝑆) = ∪ {(𝐹‘𝑦)})
12		fvex 6853	. . . . 5 ⊢ (𝐹‘𝑦) ∈ V
13	12	unisn 4886	. . . 4 ⊢ ∪ {(𝐹‘𝑦)} = (𝐹‘𝑦)
14	11, 13	eqtrdi 2780	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) ∧ 𝑦 ∈ 𝑆) → ∪ (𝐹 “ 𝑆) = (𝐹‘𝑦))
15		dffn3 6682	. . . . . 6 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)
16	15	biimpi 216	. . . . 5 ⊢ (𝐹 Fn 𝐴 → 𝐹:𝐴⟶ran 𝐹)
17	16	ad2antrr 726	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) ∧ 𝑦 ∈ 𝑆) → 𝐹:𝐴⟶ran 𝐹)
18	1	elsetpreimafvssdm 47380	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) → 𝑆 ⊆ 𝐴)
19	18	sselda 3943	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ 𝐴)
20	17, 19	ffvelcdmd 7039	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) ∧ 𝑦 ∈ 𝑆) → (𝐹‘𝑦) ∈ ran 𝐹)
21	14, 20	eqeltrd 2828	. 2 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) ∧ 𝑦 ∈ 𝑆) → ∪ (𝐹 “ 𝑆) ∈ ran 𝐹)
22	8, 21	exlimddv 1935	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) → ∪ (𝐹 “ 𝑆) ∈ ran 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2707   ≠ wne 2925   ∉ wnel 3029  ∃wrex 3053  ∅c0 4292  {csn 4585  ∪ cuni 4867  ^◡ccnv 5630  ran crn 5632   “ cima 5634   Fn wfn 6494  ⟶wf 6495  ‘cfv 6499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-fv 6507

This theorem is referenced by:  imasetpreimafvbijlemf  47395  fundcmpsurbijinjpreimafv  47401