Theorem uniimaelsetpreimafv 46799

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 46793	. . . 4 ⊢ (𝐹 Fn 𝐴 → ∅ ∉ 𝑃)
3		elnelne2 3048	. . . . . 6 ⊢ ((𝑆 ∈ 𝑃 ∧ ∅ ∉ 𝑃) → 𝑆 ≠ ∅)
4		n0 4347	. . . . . 6 ⊢ (𝑆 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝑆)
5	3, 4	sylib 217	. . . . 5 ⊢ ((𝑆 ∈ 𝑃 ∧ ∅ ∉ 𝑃) → ∃𝑦 𝑦 ∈ 𝑆)
6	5	expcom 412	. . . 4 ⊢ (∅ ∉ 𝑃 → (𝑆 ∈ 𝑃 → ∃𝑦 𝑦 ∈ 𝑆))
7	2, 6	syl 17	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝑆 ∈ 𝑃 → ∃𝑦 𝑦 ∈ 𝑆))
8	7	imp 405	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) → ∃𝑦 𝑦 ∈ 𝑆)
9	1	imaelsetpreimafv 46798	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑦 ∈ 𝑆) → (𝐹 “ 𝑆) = {(𝐹‘𝑦)})
10	9	3expa 1115	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) ∧ 𝑦 ∈ 𝑆) → (𝐹 “ 𝑆) = {(𝐹‘𝑦)})
11	10	unieqd 4921	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) ∧ 𝑦 ∈ 𝑆) → ∪ (𝐹 “ 𝑆) = ∪ {(𝐹‘𝑦)})
12		fvex 6907	. . . . 5 ⊢ (𝐹‘𝑦) ∈ V
13	12	unisn 4929	. . . 4 ⊢ ∪ {(𝐹‘𝑦)} = (𝐹‘𝑦)
14	11, 13	eqtrdi 2781	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) ∧ 𝑦 ∈ 𝑆) → ∪ (𝐹 “ 𝑆) = (𝐹‘𝑦))
15		dffn3 6733	. . . . . 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 46789	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) → 𝑆 ⊆ 𝐴)
19	18	sselda 3977	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ 𝐴)
20	17, 19	ffvelcdmd 7092	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) ∧ 𝑦 ∈ 𝑆) → (𝐹‘𝑦) ∈ ran 𝐹)
21	14, 20	eqeltrd 2825	. 2 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) ∧ 𝑦 ∈ 𝑆) → ∪ (𝐹 “ 𝑆) ∈ ran 𝐹)
22	8, 21	exlimddv 1930	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) → ∪ (𝐹 “ 𝑆) ∈ ran 𝐹)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533  ∃wex 1773   ∈ wcel 2098  {cab 2702   ≠ wne 2930   ∉ wnel 3036  ∃wrex 3060  ∅c0 4323  {csn 4629  ∪ cuni 4908  ^◡ccnv 5676  ran crn 5678   “ cima 5680   Fn wfn 6542  ⟶wf 6543  ‘cfv 6547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2696  ax-sep 5299  ax-nul 5306  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6499  df-fun 6549  df-fn 6550  df-f 6551  df-fv 6555

This theorem is referenced by:  imasetpreimafvbijlemf  46804  fundcmpsurbijinjpreimafv  46810