Theorem founiiun0 39377

Description: Union expressed as an indexed union, when a map onto is given. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Assertion

Ref	Expression
founiiun0	⊢ (𝐹:𝐴–onto→(𝐵 ∪ {∅}) → ∪ 𝐵 = ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹

Proof of Theorem founiiun0

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		uniiun 4573	. . 3 ⊢ ∪ 𝐵 = ∪ 𝑦 ∈ 𝐵 𝑦
2	1	a1i 11	. 2 ⊢ (𝐹:𝐴–onto→(𝐵 ∪ {∅}) → ∪ 𝐵 = ∪ 𝑦 ∈ 𝐵 𝑦)
3		simpl 473	. . . . . . 7 ⊢ ((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ 𝑦 ∈ 𝐵) → 𝐹:𝐴–onto→(𝐵 ∪ {∅}))
4		elun1 3780	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → 𝑦 ∈ (𝐵 ∪ {∅}))
5	4	adantl 482	. . . . . . 7 ⊢ ((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ (𝐵 ∪ {∅}))
6		foelrni 6244	. . . . . . 7 ⊢ ((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ 𝑦 ∈ (𝐵 ∪ {∅})) → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦)
7	3, 5, 6	syl2anc 693	. . . . . 6 ⊢ ((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦)
8		eqimss2 3658	. . . . . . 7 ⊢ ((𝐹‘𝑥) = 𝑦 → 𝑦 ⊆ (𝐹‘𝑥))
9	8	reximi 3011	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦 → ∃𝑥 ∈ 𝐴 𝑦 ⊆ (𝐹‘𝑥))
10	7, 9	syl 17	. . . . 5 ⊢ ((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 ⊆ (𝐹‘𝑥))
11	10	ralrimiva 2966	. . . 4 ⊢ (𝐹:𝐴–onto→(𝐵 ∪ {∅}) → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 ⊆ (𝐹‘𝑥))
12		iunss2 4565	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 ⊆ (𝐹‘𝑥) → ∪ 𝑦 ∈ 𝐵 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥))
13	11, 12	syl 17	. . 3 ⊢ (𝐹:𝐴–onto→(𝐵 ∪ {∅}) → ∪ 𝑦 ∈ 𝐵 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥))
14		simpl 473	. . . . . 6 ⊢ ((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ 𝐵 = ∅) → 𝐹:𝐴–onto→(𝐵 ∪ {∅}))
15		uneq1 3760	. . . . . . . . 9 ⊢ (𝐵 = ∅ → (𝐵 ∪ {∅}) = (∅ ∪ {∅}))
16		0un 39215	. . . . . . . . . 10 ⊢ (∅ ∪ {∅}) = {∅}
17	16	a1i 11	. . . . . . . . 9 ⊢ (𝐵 = ∅ → (∅ ∪ {∅}) = {∅})
18	15, 17	eqtrd 2656	. . . . . . . 8 ⊢ (𝐵 = ∅ → (𝐵 ∪ {∅}) = {∅})
19	18	adantl 482	. . . . . . 7 ⊢ ((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ 𝐵 = ∅) → (𝐵 ∪ {∅}) = {∅})
20		foeq3 6113	. . . . . . 7 ⊢ ((𝐵 ∪ {∅}) = {∅} → (𝐹:𝐴–onto→(𝐵 ∪ {∅}) ↔ 𝐹:𝐴–onto→{∅}))
21	19, 20	syl 17	. . . . . 6 ⊢ ((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ 𝐵 = ∅) → (𝐹:𝐴–onto→(𝐵 ∪ {∅}) ↔ 𝐹:𝐴–onto→{∅}))
22	14, 21	mpbid 222	. . . . 5 ⊢ ((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ 𝐵 = ∅) → 𝐹:𝐴–onto→{∅})
23		unisn0 39222	. . . . . . . . 9 ⊢ ∪ {∅} = ∅
24	23	eqcomi 2631	. . . . . . . 8 ⊢ ∅ = ∪ {∅}
25	24	a1i 11	. . . . . . 7 ⊢ (𝐹:𝐴–onto→{∅} → ∅ = ∪ {∅})
26		founiiun 39360	. . . . . . 7 ⊢ (𝐹:𝐴–onto→{∅} → ∪ {∅} = ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥))
27	25, 26	eqtr2d 2657	. . . . . 6 ⊢ (𝐹:𝐴–onto→{∅} → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∅)
28		0ss 3972	. . . . . . 7 ⊢ ∅ ⊆ ∪ 𝑦 ∈ 𝐵 𝑦
29	28	a1i 11	. . . . . 6 ⊢ (𝐹:𝐴–onto→{∅} → ∅ ⊆ ∪ 𝑦 ∈ 𝐵 𝑦)
30	27, 29	eqsstrd 3639	. . . . 5 ⊢ (𝐹:𝐴–onto→{∅} → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) ⊆ ∪ 𝑦 ∈ 𝐵 𝑦)
31	22, 30	syl 17	. . . 4 ⊢ ((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ 𝐵 = ∅) → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) ⊆ ∪ 𝑦 ∈ 𝐵 𝑦)
32		ssid 3624	. . . . . . . . 9 ⊢ (𝐹‘𝑥) ⊆ (𝐹‘𝑥)
33		sseq2 3627	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑥) → ((𝐹‘𝑥) ⊆ 𝑦 ↔ (𝐹‘𝑥) ⊆ (𝐹‘𝑥)))
34	33	rspcev 3309	. . . . . . . . 9 ⊢ (((𝐹‘𝑥) ∈ 𝐵 ∧ (𝐹‘𝑥) ⊆ (𝐹‘𝑥)) → ∃𝑦 ∈ 𝐵 (𝐹‘𝑥) ⊆ 𝑦)
35	32, 34	mpan2 707	. . . . . . . 8 ⊢ ((𝐹‘𝑥) ∈ 𝐵 → ∃𝑦 ∈ 𝐵 (𝐹‘𝑥) ⊆ 𝑦)
36	35	adantl 482	. . . . . . 7 ⊢ ((((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) ∧ 𝑥 ∈ 𝐴) ∧ (𝐹‘𝑥) ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝐹‘𝑥) ⊆ 𝑦)
37		simpl 473	. . . . . . . 8 ⊢ ((((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) ∧ 𝑥 ∈ 𝐴) ∧ ¬ (𝐹‘𝑥) ∈ 𝐵) → ((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) ∧ 𝑥 ∈ 𝐴))
38		fof 6115	. . . . . . . . . . . . 13 ⊢ (𝐹:𝐴–onto→(𝐵 ∪ {∅}) → 𝐹:𝐴⟶(𝐵 ∪ {∅}))
39	38	ffvelrnda 6359	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (𝐵 ∪ {∅}))
40	39	adantr 481	. . . . . . . . . . 11 ⊢ (((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ 𝑥 ∈ 𝐴) ∧ ¬ (𝐹‘𝑥) ∈ 𝐵) → (𝐹‘𝑥) ∈ (𝐵 ∪ {∅}))
41		simpr 477	. . . . . . . . . . 11 ⊢ (((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ 𝑥 ∈ 𝐴) ∧ ¬ (𝐹‘𝑥) ∈ 𝐵) → ¬ (𝐹‘𝑥) ∈ 𝐵)
42		elunnel1 3754	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑥) ∈ (𝐵 ∪ {∅}) ∧ ¬ (𝐹‘𝑥) ∈ 𝐵) → (𝐹‘𝑥) ∈ {∅})
43	40, 41, 42	syl2anc 693	. . . . . . . . . 10 ⊢ (((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ 𝑥 ∈ 𝐴) ∧ ¬ (𝐹‘𝑥) ∈ 𝐵) → (𝐹‘𝑥) ∈ {∅})
44		elsni 4194	. . . . . . . . . 10 ⊢ ((𝐹‘𝑥) ∈ {∅} → (𝐹‘𝑥) = ∅)
45	43, 44	syl 17	. . . . . . . . 9 ⊢ (((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ 𝑥 ∈ 𝐴) ∧ ¬ (𝐹‘𝑥) ∈ 𝐵) → (𝐹‘𝑥) = ∅)
46	45	adantllr 755	. . . . . . . 8 ⊢ ((((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) ∧ 𝑥 ∈ 𝐴) ∧ ¬ (𝐹‘𝑥) ∈ 𝐵) → (𝐹‘𝑥) = ∅)
47		neq0 3930	. . . . . . . . . . . . 13 ⊢ (¬ 𝐵 = ∅ ↔ ∃𝑦 𝑦 ∈ 𝐵)
48	47	biimpi 206	. . . . . . . . . . . 12 ⊢ (¬ 𝐵 = ∅ → ∃𝑦 𝑦 ∈ 𝐵)
49	48	adantr 481	. . . . . . . . . . 11 ⊢ ((¬ 𝐵 = ∅ ∧ (𝐹‘𝑥) = ∅) → ∃𝑦 𝑦 ∈ 𝐵)
50		simpr 477	. . . . . . . . . . . . . . 15 ⊢ (((𝐹‘𝑥) = ∅ ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
51		id 22	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹‘𝑥) = ∅ → (𝐹‘𝑥) = ∅)
52		0ss 3972	. . . . . . . . . . . . . . . . . 18 ⊢ ∅ ⊆ 𝑦
53	52	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹‘𝑥) = ∅ → ∅ ⊆ 𝑦)
54	51, 53	eqsstrd 3639	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑥) = ∅ → (𝐹‘𝑥) ⊆ 𝑦)
55	54	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝐹‘𝑥) = ∅ ∧ 𝑦 ∈ 𝐵) → (𝐹‘𝑥) ⊆ 𝑦)
56	50, 55	jca 554	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑥) = ∅ ∧ 𝑦 ∈ 𝐵) → (𝑦 ∈ 𝐵 ∧ (𝐹‘𝑥) ⊆ 𝑦))
57	56	ex 450	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑥) = ∅ → (𝑦 ∈ 𝐵 → (𝑦 ∈ 𝐵 ∧ (𝐹‘𝑥) ⊆ 𝑦)))
58	57	adantl 482	. . . . . . . . . . . 12 ⊢ ((¬ 𝐵 = ∅ ∧ (𝐹‘𝑥) = ∅) → (𝑦 ∈ 𝐵 → (𝑦 ∈ 𝐵 ∧ (𝐹‘𝑥) ⊆ 𝑦)))
59	58	eximdv 1846	. . . . . . . . . . 11 ⊢ ((¬ 𝐵 = ∅ ∧ (𝐹‘𝑥) = ∅) → (∃𝑦 𝑦 ∈ 𝐵 → ∃𝑦(𝑦 ∈ 𝐵 ∧ (𝐹‘𝑥) ⊆ 𝑦)))
60	49, 59	mpd 15	. . . . . . . . . 10 ⊢ ((¬ 𝐵 = ∅ ∧ (𝐹‘𝑥) = ∅) → ∃𝑦(𝑦 ∈ 𝐵 ∧ (𝐹‘𝑥) ⊆ 𝑦))
61		df-rex 2918	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ 𝐵 (𝐹‘𝑥) ⊆ 𝑦 ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ (𝐹‘𝑥) ⊆ 𝑦))
62	60, 61	sylibr 224	. . . . . . . . 9 ⊢ ((¬ 𝐵 = ∅ ∧ (𝐹‘𝑥) = ∅) → ∃𝑦 ∈ 𝐵 (𝐹‘𝑥) ⊆ 𝑦)
63	62	ad4ant24 1298	. . . . . . . 8 ⊢ ((((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) ∧ 𝑥 ∈ 𝐴) ∧ (𝐹‘𝑥) = ∅) → ∃𝑦 ∈ 𝐵 (𝐹‘𝑥) ⊆ 𝑦)
64	37, 46, 63	syl2anc 693	. . . . . . 7 ⊢ ((((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) ∧ 𝑥 ∈ 𝐴) ∧ ¬ (𝐹‘𝑥) ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝐹‘𝑥) ⊆ 𝑦)
65	36, 64	pm2.61dan 832	. . . . . 6 ⊢ (((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 (𝐹‘𝑥) ⊆ 𝑦)
66	65	ralrimiva 2966	. . . . 5 ⊢ ((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐹‘𝑥) ⊆ 𝑦)
67		iunss2 4565	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐹‘𝑥) ⊆ 𝑦 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) ⊆ ∪ 𝑦 ∈ 𝐵 𝑦)
68	66, 67	syl 17	. . . 4 ⊢ ((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) ⊆ ∪ 𝑦 ∈ 𝐵 𝑦)
69	31, 68	pm2.61dan 832	. . 3 ⊢ (𝐹:𝐴–onto→(𝐵 ∪ {∅}) → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) ⊆ ∪ 𝑦 ∈ 𝐵 𝑦)
70	13, 69	eqssd 3620	. 2 ⊢ (𝐹:𝐴–onto→(𝐵 ∪ {∅}) → ∪ 𝑦 ∈ 𝐵 𝑦 = ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥))
71	2, 70	eqtrd 2656	1 ⊢ (𝐹:𝐴–onto→(𝐵 ∪ {∅}) → ∪ 𝐵 = ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913   ∪ cun 3572   ⊆ wss 3574  ∅c0 3915  {csn 4177  ∪ cuni 4436  ∪ ciun 4520  –onto→wfo 5886  ‘cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896

This theorem is referenced by:  ismeannd  40684