Theorem iunmapsn 42757

Description: The indexed union of set exponentiations to a singleton is equal to the set exponentiation of the indexed union. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Hypotheses

Ref	Expression
iunmapsn.x	⊢ Ⅎ𝑥𝜑
iunmapsn.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
iunmapsn.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊)
iunmapsn.c	⊢ (𝜑 → 𝐶 ∈ 𝑍)

Assertion

Ref	Expression
iunmapsn	⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 (𝐵 ↑m {𝐶}) = (∪ 𝑥 ∈ 𝐴 𝐵 ↑m {𝐶}))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝑉(𝑥)   𝑊(𝑥)   𝑍(𝑥)

Proof of Theorem iunmapsn

Dummy variables 𝑓 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iunmapsn.x	. . 3 ⊢ Ⅎ𝑥𝜑
2		iunmapsn.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		iunmapsn.b	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊)
4	1, 2, 3	iunmapss 42755	. 2 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 (𝐵 ↑m {𝐶}) ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m {𝐶}))
5		simpr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m {𝐶})) → 𝑓 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m {𝐶}))
6	3	ex 413	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝑊))
7	1, 6	ralrimi 3141	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑊)
8		iunexg 7806	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑊) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V)
9	2, 7, 8	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V)
10		iunmapsn.c	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ 𝑍)
11	9, 10	mapsnd 8674	. . . . . . 7 ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐴 𝐵 ↑m {𝐶}) = {𝑓 ∣ ∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}})
12	11	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m {𝐶})) → (∪ 𝑥 ∈ 𝐴 𝐵 ↑m {𝐶}) = {𝑓 ∣ ∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}})
13	5, 12	eleqtrd 2841	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m {𝐶})) → 𝑓 ∈ {𝑓 ∣ ∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}})
14		abid 2719	. . . . 5 ⊢ (𝑓 ∈ {𝑓 ∣ ∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}} ↔ ∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩})
15	13, 14	sylib 217	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m {𝐶})) → ∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩})
16		eliun 4928	. . . . . . . . . 10 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
17	16	biimpi 215	. . . . . . . . 9 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
18	17	3ad2ant2 1133	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑓 = {⟨𝐶, 𝑦⟩}) → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
19		nfcv 2907	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑦
20		nfiu1 4958	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵
21	19, 20	nfel 2921	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵
22		nfv 1917	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝑓 = {⟨𝐶, 𝑦⟩}
23	1, 21, 22	nf3an 1904	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑓 = {⟨𝐶, 𝑦⟩})
24		rspe 3237	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑓 = {⟨𝐶, 𝑦⟩}) → ∃𝑦 ∈ 𝐵 𝑓 = {⟨𝐶, 𝑦⟩})
25	24	ancoms 459	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 = {⟨𝐶, 𝑦⟩} ∧ 𝑦 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 𝑓 = {⟨𝐶, 𝑦⟩})
26		abid 2719	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ {𝑓 ∣ ∃𝑦 ∈ 𝐵 𝑓 = {⟨𝐶, 𝑦⟩}} ↔ ∃𝑦 ∈ 𝐵 𝑓 = {⟨𝐶, 𝑦⟩})
27	25, 26	sylibr 233	. . . . . . . . . . . . . 14 ⊢ ((𝑓 = {⟨𝐶, 𝑦⟩} ∧ 𝑦 ∈ 𝐵) → 𝑓 ∈ {𝑓 ∣ ∃𝑦 ∈ 𝐵 𝑓 = {⟨𝐶, 𝑦⟩}})
28	27	adantll 711	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 = {⟨𝐶, 𝑦⟩}) ∧ 𝑦 ∈ 𝐵) → 𝑓 ∈ {𝑓 ∣ ∃𝑦 ∈ 𝐵 𝑓 = {⟨𝐶, 𝑦⟩}})
29	28	3adant2 1130	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 = {⟨𝐶, 𝑦⟩}) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑓 ∈ {𝑓 ∣ ∃𝑦 ∈ 𝐵 𝑓 = {⟨𝐶, 𝑦⟩}})
30	10	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑍)
31	3, 30	mapsnd 8674	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 ↑m {𝐶}) = {𝑓 ∣ ∃𝑦 ∈ 𝐵 𝑓 = {⟨𝐶, 𝑦⟩}})
32	31	eqcomd 2744	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → {𝑓 ∣ ∃𝑦 ∈ 𝐵 𝑓 = {⟨𝐶, 𝑦⟩}} = (𝐵 ↑m {𝐶}))
33	32	3adant3 1131	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → {𝑓 ∣ ∃𝑦 ∈ 𝐵 𝑓 = {⟨𝐶, 𝑦⟩}} = (𝐵 ↑m {𝐶}))
34	33	3adant1r 1176	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 = {⟨𝐶, 𝑦⟩}) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → {𝑓 ∣ ∃𝑦 ∈ 𝐵 𝑓 = {⟨𝐶, 𝑦⟩}} = (𝐵 ↑m {𝐶}))
35	29, 34	eleqtrd 2841	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 = {⟨𝐶, 𝑦⟩}) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑓 ∈ (𝐵 ↑m {𝐶}))
36	35	3exp 1118	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 = {⟨𝐶, 𝑦⟩}) → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝑓 ∈ (𝐵 ↑m {𝐶}))))
37	36	3adant2 1130	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑓 = {⟨𝐶, 𝑦⟩}) → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝑓 ∈ (𝐵 ↑m {𝐶}))))
38	23, 37	reximdai 3244	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑓 = {⟨𝐶, 𝑦⟩}) → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 𝑓 ∈ (𝐵 ↑m {𝐶})))
39	18, 38	mpd 15	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑓 = {⟨𝐶, 𝑦⟩}) → ∃𝑥 ∈ 𝐴 𝑓 ∈ (𝐵 ↑m {𝐶}))
40	39	3exp 1118	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 → (𝑓 = {⟨𝐶, 𝑦⟩} → ∃𝑥 ∈ 𝐴 𝑓 ∈ (𝐵 ↑m {𝐶}))))
41	40	rexlimdv 3212	. . . . 5 ⊢ (𝜑 → (∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩} → ∃𝑥 ∈ 𝐴 𝑓 ∈ (𝐵 ↑m {𝐶})))
42	41	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m {𝐶})) → (∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩} → ∃𝑥 ∈ 𝐴 𝑓 ∈ (𝐵 ↑m {𝐶})))
43	15, 42	mpd 15	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m {𝐶})) → ∃𝑥 ∈ 𝐴 𝑓 ∈ (𝐵 ↑m {𝐶}))
44		eliun 4928	. . 3 ⊢ (𝑓 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 ↑m {𝐶}) ↔ ∃𝑥 ∈ 𝐴 𝑓 ∈ (𝐵 ↑m {𝐶}))
45	43, 44	sylibr 233	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m {𝐶})) → 𝑓 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 ↑m {𝐶}))
46	4, 45	eqelssd 3942	1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 (𝐵 ↑m {𝐶}) = (∪ 𝑥 ∈ 𝐴 𝐵 ↑m {𝐶}))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539  Ⅎwnf 1786   ∈ wcel 2106  {cab 2715  ∀wral 3064  ∃wrex 3065  Vcvv 3432  {csn 4561  ⟨cop 4567  ∪ ciun 4924  (class class class)co 7275   ↑_m cmap 8615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-map 8617

This theorem is referenced by:  ovnovollem1  44194  ovnovollem2  44195