Theorem founiiun0 45178

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 5007	. 2 ⊢ ∪ 𝐵 = ∪ 𝑦 ∈ 𝐵 𝑦
2		elun1 4133	. . . . . . 7 ⊢ (𝑦 ∈ 𝐵 → 𝑦 ∈ (𝐵 ∪ {∅}))
3		foelcdmi 6884	. . . . . . 7 ⊢ ((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ 𝑦 ∈ (𝐵 ∪ {∅})) → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦)
4	2, 3	sylan2 593	. . . . . 6 ⊢ ((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦)
5		eqimss2 3995	. . . . . . 7 ⊢ ((𝐹‘𝑥) = 𝑦 → 𝑦 ⊆ (𝐹‘𝑥))
6	5	reximi 3067	. . . . . 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 3121	. . . 4 ⊢ (𝐹:𝐴–onto→(𝐵 ∪ {∅}) → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 ⊆ (𝐹‘𝑥))
9		iunss2 4998	. . . 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 482	. . . . . 6 ⊢ ((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ 𝐵 = ∅) → 𝐹:𝐴–onto→(𝐵 ∪ {∅}))
12		uneq1 4112	. . . . . . . . 9 ⊢ (𝐵 = ∅ → (𝐵 ∪ {∅}) = (∅ ∪ {∅}))
13		0un 4347	. . . . . . . . 9 ⊢ (∅ ∪ {∅}) = {∅}
14	12, 13	eqtrdi 2780	. . . . . . . 8 ⊢ (𝐵 = ∅ → (𝐵 ∪ {∅}) = {∅})
15	14	adantl 481	. . . . . . 7 ⊢ ((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ 𝐵 = ∅) → (𝐵 ∪ {∅}) = {∅})
16		foeq3 6734	. . . . . . 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 232	. . . . 5 ⊢ ((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ 𝐵 = ∅) → 𝐹:𝐴–onto→{∅})
19		founiiun 45167	. . . . . . 7 ⊢ (𝐹:𝐴–onto→{∅} → ∪ {∅} = ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥))
20		unisn0 45042	. . . . . . 7 ⊢ ∪ {∅} = ∅
21	19, 20	eqtr3di 2779	. . . . . 6 ⊢ (𝐹:𝐴–onto→{∅} → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∅)
22		0ss 4351	. . . . . 6 ⊢ ∅ ⊆ ∪ 𝑦 ∈ 𝐵 𝑦
23	21, 22	eqsstrdi 3980	. . . . 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 3958	. . . . . . . . 9 ⊢ (𝐹‘𝑥) ⊆ (𝐹‘𝑥)
26		sseq2 3962	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑥) → ((𝐹‘𝑥) ⊆ 𝑦 ↔ (𝐹‘𝑥) ⊆ (𝐹‘𝑥)))
27	26	rspcev 3577	. . . . . . . . 9 ⊢ (((𝐹‘𝑥) ∈ 𝐵 ∧ (𝐹‘𝑥) ⊆ (𝐹‘𝑥)) → ∃𝑦 ∈ 𝐵 (𝐹‘𝑥) ⊆ 𝑦)
28	25, 27	mpan2 691	. . . . . . . 8 ⊢ ((𝐹‘𝑥) ∈ 𝐵 → ∃𝑦 ∈ 𝐵 (𝐹‘𝑥) ⊆ 𝑦)
29	28	adantl 481	. . . . . . 7 ⊢ ((((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) ∧ 𝑥 ∈ 𝐴) ∧ (𝐹‘𝑥) ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝐹‘𝑥) ⊆ 𝑦)
30		fof 6736	. . . . . . . . . . . 12 ⊢ (𝐹:𝐴–onto→(𝐵 ∪ {∅}) → 𝐹:𝐴⟶(𝐵 ∪ {∅}))
31	30	ffvelcdmda 7018	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (𝐵 ∪ {∅}))
32		elunnel1 4105	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑥) ∈ (𝐵 ∪ {∅}) ∧ ¬ (𝐹‘𝑥) ∈ 𝐵) → (𝐹‘𝑥) ∈ {∅})
33	31, 32	sylan 580	. . . . . . . . . 10 ⊢ (((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ 𝑥 ∈ 𝐴) ∧ ¬ (𝐹‘𝑥) ∈ 𝐵) → (𝐹‘𝑥) ∈ {∅})
34		elsni 4594	. . . . . . . . . 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 719	. . . . . . . 8 ⊢ ((((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) ∧ 𝑥 ∈ 𝐴) ∧ ¬ (𝐹‘𝑥) ∈ 𝐵) → (𝐹‘𝑥) = ∅)
37		neq0 4303	. . . . . . . . . . . . 13 ⊢ (¬ 𝐵 = ∅ ↔ ∃𝑦 𝑦 ∈ 𝐵)
38	37	biimpi 216	. . . . . . . . . . . 12 ⊢ (¬ 𝐵 = ∅ → ∃𝑦 𝑦 ∈ 𝐵)
39	38	adantr 480	. . . . . . . . . . 11 ⊢ ((¬ 𝐵 = ∅ ∧ (𝐹‘𝑥) = ∅) → ∃𝑦 𝑦 ∈ 𝐵)
40		id 22	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑥) = ∅ → (𝐹‘𝑥) = ∅)
41		0ss 4351	. . . . . . . . . . . . . . . 16 ⊢ ∅ ⊆ 𝑦
42	40, 41	eqsstrdi 3980	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑥) = ∅ → (𝐹‘𝑥) ⊆ 𝑦)
43	42	anim1ci 616	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑥) = ∅ ∧ 𝑦 ∈ 𝐵) → (𝑦 ∈ 𝐵 ∧ (𝐹‘𝑥) ⊆ 𝑦))
44	43	ex 412	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑥) = ∅ → (𝑦 ∈ 𝐵 → (𝑦 ∈ 𝐵 ∧ (𝐹‘𝑥) ⊆ 𝑦)))
45	44	adantl 481	. . . . . . . . . . . 12 ⊢ ((¬ 𝐵 = ∅ ∧ (𝐹‘𝑥) = ∅) → (𝑦 ∈ 𝐵 → (𝑦 ∈ 𝐵 ∧ (𝐹‘𝑥) ⊆ 𝑦)))
46	45	eximdv 1917	. . . . . . . . . . 11 ⊢ ((¬ 𝐵 = ∅ ∧ (𝐹‘𝑥) = ∅) → (∃𝑦 𝑦 ∈ 𝐵 → ∃𝑦(𝑦 ∈ 𝐵 ∧ (𝐹‘𝑥) ⊆ 𝑦)))
47	39, 46	mpd 15	. . . . . . . . . 10 ⊢ ((¬ 𝐵 = ∅ ∧ (𝐹‘𝑥) = ∅) → ∃𝑦(𝑦 ∈ 𝐵 ∧ (𝐹‘𝑥) ⊆ 𝑦))
48		df-rex 3054	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ 𝐵 (𝐹‘𝑥) ⊆ 𝑦 ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ (𝐹‘𝑥) ⊆ 𝑦))
49	47, 48	sylibr 234	. . . . . . . . 9 ⊢ ((¬ 𝐵 = ∅ ∧ (𝐹‘𝑥) = ∅) → ∃𝑦 ∈ 𝐵 (𝐹‘𝑥) ⊆ 𝑦)
50	49	ad4ant24 754	. . . . . . . 8 ⊢ ((((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) ∧ 𝑥 ∈ 𝐴) ∧ (𝐹‘𝑥) = ∅) → ∃𝑦 ∈ 𝐵 (𝐹‘𝑥) ⊆ 𝑦)
51	36, 50	syldan 591	. . . . . . 7 ⊢ ((((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) ∧ 𝑥 ∈ 𝐴) ∧ ¬ (𝐹‘𝑥) ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝐹‘𝑥) ⊆ 𝑦)
52	29, 51	pm2.61dan 812	. . . . . 6 ⊢ (((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 (𝐹‘𝑥) ⊆ 𝑦)
53	52	ralrimiva 3121	. . . . 5 ⊢ ((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐹‘𝑥) ⊆ 𝑦)
54		iunss2 4998	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐹‘𝑥) ⊆ 𝑦 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) ⊆ ∪ 𝑦 ∈ 𝐵 𝑦)
55	53, 54	syl 17	. . . 4 ⊢ ((𝐹:𝐴–onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) ⊆ ∪ 𝑦 ∈ 𝐵 𝑦)
56	24, 55	pm2.61dan 812	. . 3 ⊢ (𝐹:𝐴–onto→(𝐵 ∪ {∅}) → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) ⊆ ∪ 𝑦 ∈ 𝐵 𝑦)
57	10, 56	eqssd 3953	. 2 ⊢ (𝐹:𝐴–onto→(𝐵 ∪ {∅}) → ∪ 𝑦 ∈ 𝐵 𝑦 = ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥))
58	1, 57	eqtrid 2776	1 ⊢ (𝐹:𝐴–onto→(𝐵 ∪ {∅}) → ∪ 𝐵 = ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   ∪ cun 3901   ⊆ wss 3903  ∅c0 4284  {csn 4577  ∪ cuni 4858  ∪ ciun 4941  –onto→wfo 6480  ‘cfv 6482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-fo 6488  df-fv 6490

This theorem is referenced by:  ismeannd  46458