Theorem founiiun0 45769

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 5017	. 2 ⊢ ∪ 𝐵 = ∪ 𝑦 ∈ 𝐵 𝑦
2		elun1 4135	. . . . . . 7 ⊢ (𝑦 ∈ 𝐵 → 𝑦 ∈ (𝐵 ∪ {∅}))
3		foelcdmi 6929	. . . . . . 7 ⊢ ((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ 𝑦 ∈ (𝐵 ∪ {∅})) → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦)
4	2, 3	sylan2 602	. . . . . 6 ⊢ ((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦)
5		eqimss2 3996	. . . . . . 7 ⊢ ((𝐹‘𝑥) = 𝑦 → 𝑦 ⊆ (𝐹‘𝑥))
6	5	reximi 3101	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦 → ∃𝑥 ∈ 𝐴 𝑦 ⊆ (𝐹‘𝑥))
7	4, 6	syl 17	. . . . 5 ⊢ ((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 ⊆ (𝐹‘𝑥))
8	7	ralrimiva 3155	. . . 4 ⊢ (𝐹:𝐴–onto→(𝐵 ∪ {∅}) → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 ⊆ (𝐹‘𝑥))
9		iunss2 5008	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 ⊆ (𝐹‘𝑥) → ∪ 𝑦 ∈ 𝐵 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥))
10	8, 9	syl 17	. . 3 ⊢ (𝐹:𝐴–onto→(𝐵 ∪ {∅}) → ∪ 𝑦 ∈ 𝐵 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥))
11		simpl 486	. . . . . 6 ⊢ ((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ 𝐵 = ∅) → 𝐹:𝐴–onto→(𝐵 ∪ {∅}))
12		uneq1 4115	. . . . . . . . 9 ⊢ (𝐵 = ∅ → (𝐵 ∪ {∅}) = (∅ ∪ {∅}))
13		0un 4351	. . . . . . . . 9 ⊢ (∅ ∪ {∅}) = {∅}
14	12, 13	eqtrdi 2814	. . . . . . . 8 ⊢ (𝐵 = ∅ → (𝐵 ∪ {∅}) = {∅})
15	14	adantl 485	. . . . . . 7 ⊢ ((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ 𝐵 = ∅) → (𝐵 ∪ {∅}) = {∅})
16		foeq3 6777	. . . . . . 7 ⊢ ((𝐵 ∪ {∅}) = {∅} → (𝐹:𝐴–onto→(𝐵 ∪ {∅}) ↔ 𝐹:𝐴–onto→{∅}))
17	15, 16	syl 17	. . . . . 6 ⊢ ((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ 𝐵 = ∅) → (𝐹:𝐴–onto→(𝐵 ∪ {∅}) ↔ 𝐹:𝐴–onto→{∅}))
18	11, 17	mpbid 234	. . . . 5 ⊢ ((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ 𝐵 = ∅) → 𝐹:𝐴–onto→{∅})
19		founiiun 45758	. . . . . . 7 ⊢ (𝐹:𝐴–onto→{∅} → ∪ {∅} = ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥))
20		unisn0 45635	. . . . . . 7 ⊢ ∪ {∅} = ∅
21	19, 20	eqtr3di 2813	. . . . . 6 ⊢ (𝐹:𝐴–onto→{∅} → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∅)
22		0ss 4355	. . . . . 6 ⊢ ∅ ⊆ ∪ 𝑦 ∈ 𝐵 𝑦
23	21, 22	eqsstrdi 3981	. . . . 5 ⊢ (𝐹:𝐴–onto→{∅} → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) ⊆ ∪ 𝑦 ∈ 𝐵 𝑦)
24	18, 23	syl 17	. . . 4 ⊢ ((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ 𝐵 = ∅) → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) ⊆ ∪ 𝑦 ∈ 𝐵 𝑦)
25		ssid 3959	. . . . . . . . 9 ⊢ (𝐹‘𝑥) ⊆ (𝐹‘𝑥)
26		sseq2 3963	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑥) → ((𝐹‘𝑥) ⊆ 𝑦 ↔ (𝐹‘𝑥) ⊆ (𝐹‘𝑥)))
27	26	rspcev 3582	. . . . . . . . 9 ⊢ (((𝐹‘𝑥) ∈ 𝐵 ∧ (𝐹‘𝑥) ⊆ (𝐹‘𝑥)) → ∃𝑦 ∈ 𝐵 (𝐹‘𝑥) ⊆ 𝑦)
28	25, 27	mpan2 701	. . . . . . . 8 ⊢ ((𝐹‘𝑥) ∈ 𝐵 → ∃𝑦 ∈ 𝐵 (𝐹‘𝑥) ⊆ 𝑦)
29	28	adantl 485	. . . . . . 7 ⊢ ((((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) ∧ 𝑥 ∈ 𝐴) ∧ (𝐹‘𝑥) ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝐹‘𝑥) ⊆ 𝑦)
30		fof 6779	. . . . . . . . . . . 12 ⊢ (𝐹:𝐴–onto→(𝐵 ∪ {∅}) → 𝐹:𝐴⟶(𝐵 ∪ {∅}))
31	30	ffvelcdmda 7066	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (𝐵 ∪ {∅}))
32		elunnel1 4108	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑥) ∈ (𝐵 ∪ {∅}) ∧ ¬ (𝐹‘𝑥) ∈ 𝐵) → (𝐹‘𝑥) ∈ {∅})
33	31, 32	sylan 589	. . . . . . . . . 10 ⊢ (((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ 𝑥 ∈ 𝐴) ∧ ¬ (𝐹‘𝑥) ∈ 𝐵) → (𝐹‘𝑥) ∈ {∅})
34		elsni 4600	. . . . . . . . . 10 ⊢ ((𝐹‘𝑥) ∈ {∅} → (𝐹‘𝑥) = ∅)
35	33, 34	syl 17	. . . . . . . . 9 ⊢ (((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ 𝑥 ∈ 𝐴) ∧ ¬ (𝐹‘𝑥) ∈ 𝐵) → (𝐹‘𝑥) = ∅)
36	35	adantllr 729	. . . . . . . 8 ⊢ ((((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) ∧ 𝑥 ∈ 𝐴) ∧ ¬ (𝐹‘𝑥) ∈ 𝐵) → (𝐹‘𝑥) = ∅)
37		neq0 4305	. . . . . . . . . . . 12 ⊢ (¬ 𝐵 = ∅ ↔ ∃𝑦 𝑦 ∈ 𝐵)
38	37	birani 507	. . . . . . . . . . 11 ⊢ ((¬ 𝐵 = ∅ ∧ (𝐹‘𝑥) = ∅) → ∃𝑦 𝑦 ∈ 𝐵)
39		id 22	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑥) = ∅ → (𝐹‘𝑥) = ∅)
40		0ss 4355	. . . . . . . . . . . . . . . 16 ⊢ ∅ ⊆ 𝑦
41	39, 40	eqsstrdi 3981	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑥) = ∅ → (𝐹‘𝑥) ⊆ 𝑦)
42	41	anim1ci 625	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑥) = ∅ ∧ 𝑦 ∈ 𝐵) → (𝑦 ∈ 𝐵 ∧ (𝐹‘𝑥) ⊆ 𝑦))
43	42	ex 416	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑥) = ∅ → (𝑦 ∈ 𝐵 → (𝑦 ∈ 𝐵 ∧ (𝐹‘𝑥) ⊆ 𝑦)))
44	43	adantl 485	. . . . . . . . . . . 12 ⊢ ((¬ 𝐵 = ∅ ∧ (𝐹‘𝑥) = ∅) → (𝑦 ∈ 𝐵 → (𝑦 ∈ 𝐵 ∧ (𝐹‘𝑥) ⊆ 𝑦)))
45	44	eximdv 1938	. . . . . . . . . . 11 ⊢ ((¬ 𝐵 = ∅ ∧ (𝐹‘𝑥) = ∅) → (∃𝑦 𝑦 ∈ 𝐵 → ∃𝑦(𝑦 ∈ 𝐵 ∧ (𝐹‘𝑥) ⊆ 𝑦)))
46	38, 45	mpd 15	. . . . . . . . . 10 ⊢ ((¬ 𝐵 = ∅ ∧ (𝐹‘𝑥) = ∅) → ∃𝑦(𝑦 ∈ 𝐵 ∧ (𝐹‘𝑥) ⊆ 𝑦))
47		df-rex 3088	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ 𝐵 (𝐹‘𝑥) ⊆ 𝑦 ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ (𝐹‘𝑥) ⊆ 𝑦))
48	46, 47	sylibr 236	. . . . . . . . 9 ⊢ ((¬ 𝐵 = ∅ ∧ (𝐹‘𝑥) = ∅) → ∃𝑦 ∈ 𝐵 (𝐹‘𝑥) ⊆ 𝑦)
49	48	ad4ant24 764	. . . . . . . 8 ⊢ ((((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) ∧ 𝑥 ∈ 𝐴) ∧ (𝐹‘𝑥) = ∅) → ∃𝑦 ∈ 𝐵 (𝐹‘𝑥) ⊆ 𝑦)
50	36, 49	syldan 600	. . . . . . 7 ⊢ ((((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) ∧ 𝑥 ∈ 𝐴) ∧ ¬ (𝐹‘𝑥) ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝐹‘𝑥) ⊆ 𝑦)
51	29, 50	pm2.61dan 822	. . . . . 6 ⊢ (((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 (𝐹‘𝑥) ⊆ 𝑦)
52	51	ralrimiva 3155	. . . . 5 ⊢ ((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐹‘𝑥) ⊆ 𝑦)
53		iunss2 5008	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐹‘𝑥) ⊆ 𝑦 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) ⊆ ∪ 𝑦 ∈ 𝐵 𝑦)
54	52, 53	syl 17	. . . 4 ⊢ ((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) ⊆ ∪ 𝑦 ∈ 𝐵 𝑦)
55	24, 54	pm2.61dan 822	. . 3 ⊢ (𝐹:𝐴–onto→(𝐵 ∪ {∅}) → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) ⊆ ∪ 𝑦 ∈ 𝐵 𝑦)
56	10, 55	eqssd 3954	. 2 ⊢ (𝐹:𝐴–onto→(𝐵 ∪ {∅}) → ∪ 𝑦 ∈ 𝐵 𝑦 = ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥))
57	1, 56	eqtrid 2810	1 ⊢ (𝐹:𝐴–onto→(𝐵 ∪ {∅}) → ∪ 𝐵 = ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1561  ∃wex 1800   ∈ wcel 2143  ∀wral 3077  ∃wrex 3087   ∪ cun 3903   ⊆ wss 3905  ∅c0 4286  {csn 4583  ∪ cuni 4866  ∪ ciun 4950  –onto→wfo 6520  ‘cfv 6522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-nul 5257  ax-pr 5391

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-fo 6528  df-fv 6530

This theorem is referenced by:  ismeannd  47042