Theorem iunmapsn 38404

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 38402	. 2 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 (𝐵 ↑𝑚 {𝐶}) ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑𝑚 {𝐶}))
5		simpr 476	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 ↑𝑚 {𝐶})) → 𝑓 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 ↑𝑚 {𝐶}))
6	3	ex 449	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝑊))
7	1, 6	ralrimi 2940	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑊)
8		iunexg 7035	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑊) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V)
9	2, 7, 8	syl2anc 691	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V)
10		iunmapsn.c	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ 𝑍)
11	9, 10	mapsnd 38383	. . . . . . . . 9 ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐴 𝐵 ↑𝑚 {𝐶}) = {𝑓 ∣ ∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}})
12	11	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 ↑𝑚 {𝐶})) → (∪ 𝑥 ∈ 𝐴 𝐵 ↑𝑚 {𝐶}) = {𝑓 ∣ ∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}})
13	5, 12	eleqtrd 2690	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 ↑𝑚 {𝐶})) → 𝑓 ∈ {𝑓 ∣ ∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}})
14		abid 2598	. . . . . . 7 ⊢ (𝑓 ∈ {𝑓 ∣ ∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}} ↔ ∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩})
15	13, 14	sylib 207	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 ↑𝑚 {𝐶})) → ∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩})
16		eliun 4460	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
17	16	biimpi 205	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
18	17	3ad2ant2 1076	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑓 = {⟨𝐶, 𝑦⟩}) → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
19		nfcv 2751	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥𝑦
20		nfiu1 4486	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵
21	19, 20	nfel 2763	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵
22		nfv 1830	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥 𝑓 = {⟨𝐶, 𝑦⟩}
23	1, 21, 22	nf3an 1819	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑓 = {⟨𝐶, 𝑦⟩})
24		rspe 2986	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑓 = {⟨𝐶, 𝑦⟩}) → ∃𝑦 ∈ 𝐵 𝑓 = {⟨𝐶, 𝑦⟩})
25	24	ancoms 468	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓 = {⟨𝐶, 𝑦⟩} ∧ 𝑦 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 𝑓 = {⟨𝐶, 𝑦⟩})
26		abid 2598	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ {𝑓 ∣ ∃𝑦 ∈ 𝐵 𝑓 = {⟨𝐶, 𝑦⟩}} ↔ ∃𝑦 ∈ 𝐵 𝑓 = {⟨𝐶, 𝑦⟩})
27	25, 26	sylibr 223	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 = {⟨𝐶, 𝑦⟩} ∧ 𝑦 ∈ 𝐵) → 𝑓 ∈ {𝑓 ∣ ∃𝑦 ∈ 𝐵 𝑓 = {⟨𝐶, 𝑦⟩}})
28	27	adantll 746	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑓 = {⟨𝐶, 𝑦⟩}) ∧ 𝑦 ∈ 𝐵) → 𝑓 ∈ {𝑓 ∣ ∃𝑦 ∈ 𝐵 𝑓 = {⟨𝐶, 𝑦⟩}})
29	28	3adant2 1073	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓 = {⟨𝐶, 𝑦⟩}) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑓 ∈ {𝑓 ∣ ∃𝑦 ∈ 𝐵 𝑓 = {⟨𝐶, 𝑦⟩}})
30	10	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑍)
31	3, 30	mapsnd 38383	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 ↑𝑚 {𝐶}) = {𝑓 ∣ ∃𝑦 ∈ 𝐵 𝑓 = {⟨𝐶, 𝑦⟩}})
32	31	eqcomd 2616	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → {𝑓 ∣ ∃𝑦 ∈ 𝐵 𝑓 = {⟨𝐶, 𝑦⟩}} = (𝐵 ↑𝑚 {𝐶}))
33	32	3adant3 1074	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → {𝑓 ∣ ∃𝑦 ∈ 𝐵 𝑓 = {⟨𝐶, 𝑦⟩}} = (𝐵 ↑𝑚 {𝐶}))
34	33	3adant1r 1311	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓 = {⟨𝐶, 𝑦⟩}) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → {𝑓 ∣ ∃𝑦 ∈ 𝐵 𝑓 = {⟨𝐶, 𝑦⟩}} = (𝐵 ↑𝑚 {𝐶}))
35	29, 34	eleqtrd 2690	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 = {⟨𝐶, 𝑦⟩}) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑓 ∈ (𝐵 ↑𝑚 {𝐶}))
36	35	3exp 1256	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 = {⟨𝐶, 𝑦⟩}) → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝑓 ∈ (𝐵 ↑𝑚 {𝐶}))))
37	36	3adant2 1073	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑓 = {⟨𝐶, 𝑦⟩}) → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝑓 ∈ (𝐵 ↑𝑚 {𝐶}))))
38	23, 37	reximdai 2995	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑓 = {⟨𝐶, 𝑦⟩}) → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 𝑓 ∈ (𝐵 ↑𝑚 {𝐶})))
39	18, 38	mpd 15	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑓 = {⟨𝐶, 𝑦⟩}) → ∃𝑥 ∈ 𝐴 𝑓 ∈ (𝐵 ↑𝑚 {𝐶}))
40	39	3exp 1256	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 → (𝑓 = {⟨𝐶, 𝑦⟩} → ∃𝑥 ∈ 𝐴 𝑓 ∈ (𝐵 ↑𝑚 {𝐶}))))
41	40	rexlimdv 3012	. . . . . . 7 ⊢ (𝜑 → (∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩} → ∃𝑥 ∈ 𝐴 𝑓 ∈ (𝐵 ↑𝑚 {𝐶})))
42	41	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 ↑𝑚 {𝐶})) → (∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩} → ∃𝑥 ∈ 𝐴 𝑓 ∈ (𝐵 ↑𝑚 {𝐶})))
43	15, 42	mpd 15	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 ↑𝑚 {𝐶})) → ∃𝑥 ∈ 𝐴 𝑓 ∈ (𝐵 ↑𝑚 {𝐶}))
44		eliun 4460	. . . . 5 ⊢ (𝑓 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 ↑𝑚 {𝐶}) ↔ ∃𝑥 ∈ 𝐴 𝑓 ∈ (𝐵 ↑𝑚 {𝐶}))
45	43, 44	sylibr 223	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 ↑𝑚 {𝐶})) → 𝑓 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 ↑𝑚 {𝐶}))
46	45	ralrimiva 2949	. . 3 ⊢ (𝜑 → ∀𝑓 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 ↑𝑚 {𝐶})𝑓 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 ↑𝑚 {𝐶}))
47		dfss3 3558	. . 3 ⊢ ((∪ 𝑥 ∈ 𝐴 𝐵 ↑𝑚 {𝐶}) ⊆ ∪ 𝑥 ∈ 𝐴 (𝐵 ↑𝑚 {𝐶}) ↔ ∀𝑓 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 ↑𝑚 {𝐶})𝑓 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 ↑𝑚 {𝐶}))
48	46, 47	sylibr 223	. 2 ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐴 𝐵 ↑𝑚 {𝐶}) ⊆ ∪ 𝑥 ∈ 𝐴 (𝐵 ↑𝑚 {𝐶}))
49	4, 48	eqssd 3585	1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 (𝐵 ↑𝑚 {𝐶}) = (∪ 𝑥 ∈ 𝐴 𝐵 ↑𝑚 {𝐶}))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977  {cab 2596  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  {csn 4125  ⟨cop 4131  ∪ ciun 4455  (class class class)co 6549   ↑_𝑚 cmap 7744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746

This theorem is referenced by:  ovnovollem1  39546  ovnovollem2  39547