Theorem uniimaelsetpreimafv 45900

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 45894	. . . 4 ⊢ (𝐹 Fn 𝐴 → ∅ ∉ 𝑃)
3		elnelne2 3058	. . . . . 6 ⊢ ((𝑆 ∈ 𝑃 ∧ ∅ ∉ 𝑃) → 𝑆 ≠ ∅)
4		n0 4343	. . . . . 6 ⊢ (𝑆 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝑆)
5	3, 4	sylib 217	. . . . 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 45899	. . . . . 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 4916	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) ∧ 𝑦 ∈ 𝑆) → ∪ (𝐹 “ 𝑆) = ∪ {(𝐹‘𝑦)})
12		fvex 6892	. . . . 5 ⊢ (𝐹‘𝑦) ∈ V
13	12	unisn 4924	. . . 4 ⊢ ∪ {(𝐹‘𝑦)} = (𝐹‘𝑦)
14	11, 13	eqtrdi 2788	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) ∧ 𝑦 ∈ 𝑆) → ∪ (𝐹 “ 𝑆) = (𝐹‘𝑦))
15		dffn3 6718	. . . . . 6 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)
16	15	biimpi 215	. . . . 5 ⊢ (𝐹 Fn 𝐴 → 𝐹:𝐴⟶ran 𝐹)
17	16	ad2antrr 724	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) ∧ 𝑦 ∈ 𝑆) → 𝐹:𝐴⟶ran 𝐹)
18	1	elsetpreimafvssdm 45890	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) → 𝑆 ⊆ 𝐴)
19	18	sselda 3979	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ 𝐴)
20	17, 19	ffvelcdmd 7073	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) ∧ 𝑦 ∈ 𝑆) → (𝐹‘𝑦) ∈ ran 𝐹)
21	14, 20	eqeltrd 2833	. 2 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) ∧ 𝑦 ∈ 𝑆) → ∪ (𝐹 “ 𝑆) ∈ ran 𝐹)
22	8, 21	exlimddv 1938	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) → ∪ (𝐹 “ 𝑆) ∈ ran 𝐹)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106  {cab 2709   ≠ wne 2940   ∉ wnel 3046  ∃wrex 3070  ∅c0 4319  {csn 4623  ∪ cuni 4902  ^◡ccnv 5669  ran crn 5671   “ cima 5673   Fn wfn 6528  ⟶wf 6529  ‘cfv 6533

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5293  ax-nul 5300  ax-pr 5421

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5143  df-opab 5205  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541

This theorem is referenced by:  imasetpreimafvbijlemf  45905  fundcmpsurbijinjpreimafv  45911