Theorem iunmapsn 45670

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 45668	. 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 3237	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑊)
8		iunexg 7906	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑊) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V)
9	2, 7, 8	syl2anc 590	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V)
10		iunmapsn.c	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ 𝑍)
11	9, 10	mapsnd 8825	. . . . . . 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 2721	. . . . 5 ⊢ (𝑓 ∈ {𝑓 ∣ ∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}} ↔ ∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩})
15	13, 14	sylib 219	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m {𝐶})) → ∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩})
16		eliun 4926	. . . . . . . . . 10 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
17	16	biimpi 217	. . . . . . . . 9 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
18	17	3ad2ant2 1140	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑓 = {⟨𝐶, 𝑦⟩}) → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
19		nfcv 2901	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑦
20		nfiu1 4958	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵
21	19, 20	nfel 2915	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵
22		nfv 1921	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝑓 = {⟨𝐶, 𝑦⟩}
23	1, 21, 22	nf3an 1908	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑓 = {⟨𝐶, 𝑦⟩})
24		rspe 3229	. . . . . . . . . . . . . . . 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 2721	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ {𝑓 ∣ ∃𝑦 ∈ 𝐵 𝑓 = {⟨𝐶, 𝑦⟩}} ↔ ∃𝑦 ∈ 𝐵 𝑓 = {⟨𝐶, 𝑦⟩})
27	25, 26	sylibr 235	. . . . . . . . . . . . . 14 ⊢ ((𝑓 = {⟨𝐶, 𝑦⟩} ∧ 𝑦 ∈ 𝐵) → 𝑓 ∈ {𝑓 ∣ ∃𝑦 ∈ 𝐵 𝑓 = {⟨𝐶, 𝑦⟩}})
28	27	adantll 720	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 = {⟨𝐶, 𝑦⟩}) ∧ 𝑦 ∈ 𝐵) → 𝑓 ∈ {𝑓 ∣ ∃𝑦 ∈ 𝐵 𝑓 = {⟨𝐶, 𝑦⟩}})
29	28	3adant2 1137	. . . . . . . . . . . 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 8825	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 ↑m {𝐶}) = {𝑓 ∣ ∃𝑦 ∈ 𝐵 𝑓 = {⟨𝐶, 𝑦⟩}})
32	31	eqcomd 2745	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → {𝑓 ∣ ∃𝑦 ∈ 𝐵 𝑓 = {⟨𝐶, 𝑦⟩}} = (𝐵 ↑m {𝐶}))
33	32	3adant3 1138	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → {𝑓 ∣ ∃𝑦 ∈ 𝐵 𝑓 = {⟨𝐶, 𝑦⟩}} = (𝐵 ↑m {𝐶}))
34	33	3adant1r 1184	. . . . . . . . . . . 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 1125	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 = {⟨𝐶, 𝑦⟩}) → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝑓 ∈ (𝐵 ↑m {𝐶}))))
37	36	3adant2 1137	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑓 = {⟨𝐶, 𝑦⟩}) → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝑓 ∈ (𝐵 ↑m {𝐶}))))
38	23, 37	reximdai 3241	. . . . . . . 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 1125	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 → (𝑓 = {⟨𝐶, 𝑦⟩} → ∃𝑥 ∈ 𝐴 𝑓 ∈ (𝐵 ↑m {𝐶}))))
41	40	rexlimdv 3138	. . . . 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 4926	. . 3 ⊢ (𝑓 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 ↑m {𝐶}) ↔ ∃𝑥 ∈ 𝐴 𝑓 ∈ (𝐵 ↑m {𝐶}))
45	43, 44	sylibr 235	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m {𝐶})) → 𝑓 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 ↑m {𝐶}))
46	4, 45	eqelssd 3936	1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 (𝐵 ↑m {𝐶}) = (∪ 𝑥 ∈ 𝐴 𝐵 ↑m {𝐶}))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547  Ⅎwnf 1790   ∈ wcel 2119  {cab 2717  ∀wral 3053  ∃wrex 3063  Vcvv 3431  {csn 4556  ⟨cop 4562  ∪ ciun 4922  (class class class)co 7357   ↑_m cmap 8764

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-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5200  ax-sep 5219  ax-nul 5229  ax-pow 5295  ax-pr 5363  ax-un 7679

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-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-iun 4924  df-br 5074  df-opab 5136  df-mpt 5155  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-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7932  df-2nd 7933  df-map 8766

This theorem is referenced by:  ovnovollem1  47107  ovnovollem2  47108