Theorem iunmapsn 41478

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 41476	. 2 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 (𝐵 ↑m {𝐶}) ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m {𝐶}))
5		simpr 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m {𝐶})) → 𝑓 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m {𝐶}))
6	3	ex 415	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝑊))
7	1, 6	ralrimi 3216	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑊)
8		iunexg 7663	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑊) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V)
9	2, 7, 8	syl2anc 586	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V)
10		iunmapsn.c	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ 𝑍)
11	9, 10	mapsnd 8449	. . . . . . 7 ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐴 𝐵 ↑m {𝐶}) = {𝑓 ∣ ∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}})
12	11	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m {𝐶})) → (∪ 𝑥 ∈ 𝐴 𝐵 ↑m {𝐶}) = {𝑓 ∣ ∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}})
13	5, 12	eleqtrd 2915	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m {𝐶})) → 𝑓 ∈ {𝑓 ∣ ∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}})
14		abid 2803	. . . . 5 ⊢ (𝑓 ∈ {𝑓 ∣ ∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}} ↔ ∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩})
15	13, 14	sylib 220	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m {𝐶})) → ∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩})
16		eliun 4922	. . . . . . . . . 10 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
17	16	biimpi 218	. . . . . . . . 9 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
18	17	3ad2ant2 1130	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑓 = {⟨𝐶, 𝑦⟩}) → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
19		nfcv 2977	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑦
20		nfiu1 4952	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵
21	19, 20	nfel 2992	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵
22		nfv 1911	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝑓 = {⟨𝐶, 𝑦⟩}
23	1, 21, 22	nf3an 1898	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑓 = {⟨𝐶, 𝑦⟩})
24		rspe 3304	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑓 = {⟨𝐶, 𝑦⟩}) → ∃𝑦 ∈ 𝐵 𝑓 = {⟨𝐶, 𝑦⟩})
25	24	ancoms 461	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 = {⟨𝐶, 𝑦⟩} ∧ 𝑦 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 𝑓 = {⟨𝐶, 𝑦⟩})
26		abid 2803	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ {𝑓 ∣ ∃𝑦 ∈ 𝐵 𝑓 = {⟨𝐶, 𝑦⟩}} ↔ ∃𝑦 ∈ 𝐵 𝑓 = {⟨𝐶, 𝑦⟩})
27	25, 26	sylibr 236	. . . . . . . . . . . . . 14 ⊢ ((𝑓 = {⟨𝐶, 𝑦⟩} ∧ 𝑦 ∈ 𝐵) → 𝑓 ∈ {𝑓 ∣ ∃𝑦 ∈ 𝐵 𝑓 = {⟨𝐶, 𝑦⟩}})
28	27	adantll 712	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 = {⟨𝐶, 𝑦⟩}) ∧ 𝑦 ∈ 𝐵) → 𝑓 ∈ {𝑓 ∣ ∃𝑦 ∈ 𝐵 𝑓 = {⟨𝐶, 𝑦⟩}})
29	28	3adant2 1127	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 = {⟨𝐶, 𝑦⟩}) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑓 ∈ {𝑓 ∣ ∃𝑦 ∈ 𝐵 𝑓 = {⟨𝐶, 𝑦⟩}})
30	10	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑍)
31	3, 30	mapsnd 8449	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 ↑m {𝐶}) = {𝑓 ∣ ∃𝑦 ∈ 𝐵 𝑓 = {⟨𝐶, 𝑦⟩}})
32	31	eqcomd 2827	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → {𝑓 ∣ ∃𝑦 ∈ 𝐵 𝑓 = {⟨𝐶, 𝑦⟩}} = (𝐵 ↑m {𝐶}))
33	32	3adant3 1128	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → {𝑓 ∣ ∃𝑦 ∈ 𝐵 𝑓 = {⟨𝐶, 𝑦⟩}} = (𝐵 ↑m {𝐶}))
34	33	3adant1r 1173	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 = {⟨𝐶, 𝑦⟩}) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → {𝑓 ∣ ∃𝑦 ∈ 𝐵 𝑓 = {⟨𝐶, 𝑦⟩}} = (𝐵 ↑m {𝐶}))
35	29, 34	eleqtrd 2915	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 = {⟨𝐶, 𝑦⟩}) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑓 ∈ (𝐵 ↑m {𝐶}))
36	35	3exp 1115	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 = {⟨𝐶, 𝑦⟩}) → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝑓 ∈ (𝐵 ↑m {𝐶}))))
37	36	3adant2 1127	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑓 = {⟨𝐶, 𝑦⟩}) → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝑓 ∈ (𝐵 ↑m {𝐶}))))
38	23, 37	reximdai 3311	. . . . . . . 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 1115	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 → (𝑓 = {⟨𝐶, 𝑦⟩} → ∃𝑥 ∈ 𝐴 𝑓 ∈ (𝐵 ↑m {𝐶}))))
41	40	rexlimdv 3283	. . . . 5 ⊢ (𝜑 → (∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩} → ∃𝑥 ∈ 𝐴 𝑓 ∈ (𝐵 ↑m {𝐶})))
42	41	adantr 483	. . . 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 4922	. . 3 ⊢ (𝑓 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 ↑m {𝐶}) ↔ ∃𝑥 ∈ 𝐴 𝑓 ∈ (𝐵 ↑m {𝐶}))
45	43, 44	sylibr 236	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m {𝐶})) → 𝑓 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 ↑m {𝐶}))
46	4, 45	eqelssd 3987	1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 (𝐵 ↑m {𝐶}) = (∪ 𝑥 ∈ 𝐴 𝐵 ↑m {𝐶}))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1533  Ⅎwnf 1780   ∈ wcel 2110  {cab 2799  ∀wral 3138  ∃wrex 3139  Vcvv 3494  {csn 4566  ⟨cop 4572  ∪ ciun 4918  (class class class)co 7155   ↑_m cmap 8405

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7158  df-oprab 7159  df-mpo 7160  df-1st 7688  df-2nd 7689  df-map 8407

This theorem is referenced by:  ovnovollem1  42937  ovnovollem2  42938