Theorem iunfo 9240

Description: Existence of an onto function from a disjoint union to a union. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 18-Jan-2014.)

Hypothesis

Ref	Expression
iunfo.1	⊢ 𝑇 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)

Assertion

Ref	Expression
iunfo	⊢ (2nd ↾ 𝑇):𝑇–onto→∪ 𝑥 ∈ 𝐴 𝐵

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝐵(𝑥)   𝑇(𝑥)

Proof of Theorem iunfo

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fo2nd 7080	. . . 4 ⊢ 2nd :V–onto→V
2		fof 6028	. . . 4 ⊢ (2nd :V–onto→V → 2nd :V⟶V)
3		ffn 5958	. . . 4 ⊢ (2nd :V⟶V → 2nd Fn V)
4	1, 2, 3	mp2b 10	. . 3 ⊢ 2nd Fn V
5		ssv 3588	. . 3 ⊢ 𝑇 ⊆ V
6		fnssres 5918	. . 3 ⊢ ((2nd Fn V ∧ 𝑇 ⊆ V) → (2nd ↾ 𝑇) Fn 𝑇)
7	4, 5, 6	mp2an 704	. 2 ⊢ (2nd ↾ 𝑇) Fn 𝑇
8		df-ima 5051	. . 3 ⊢ (2nd “ 𝑇) = ran (2nd ↾ 𝑇)
9		iunfo.1	. . . . . . . . . . 11 ⊢ 𝑇 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
10	9	eleq2i 2680	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑇 ↔ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))
11		eliun 4460	. . . . . . . . . 10 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ ({𝑥} × 𝐵))
12	10, 11	bitri 263	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑇 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ ({𝑥} × 𝐵))
13		xp2nd 7090	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ({𝑥} × 𝐵) → (2nd ‘𝑧) ∈ 𝐵)
14		eleq1 2676	. . . . . . . . . . 11 ⊢ ((2nd ‘𝑧) = 𝑦 → ((2nd ‘𝑧) ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
15	13, 14	syl5ib 233	. . . . . . . . . 10 ⊢ ((2nd ‘𝑧) = 𝑦 → (𝑧 ∈ ({𝑥} × 𝐵) → 𝑦 ∈ 𝐵))
16	15	reximdv 2999	. . . . . . . . 9 ⊢ ((2nd ‘𝑧) = 𝑦 → (∃𝑥 ∈ 𝐴 𝑧 ∈ ({𝑥} × 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵))
17	12, 16	syl5bi 231	. . . . . . . 8 ⊢ ((2nd ‘𝑧) = 𝑦 → (𝑧 ∈ 𝑇 → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵))
18	17	impcom 445	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑇 ∧ (2nd ‘𝑧) = 𝑦) → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
19	18	rexlimiva 3010	. . . . . 6 ⊢ (∃𝑧 ∈ 𝑇 (2nd ‘𝑧) = 𝑦 → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
20		nfiu1 4486	. . . . . . . . 9 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
21	9, 20	nfcxfr 2749	. . . . . . . 8 ⊢ Ⅎ𝑥𝑇
22		nfv 1830	. . . . . . . 8 ⊢ Ⅎ𝑥(2nd ‘𝑧) = 𝑦
23	21, 22	nfrex 2990	. . . . . . 7 ⊢ Ⅎ𝑥∃𝑧 ∈ 𝑇 (2nd ‘𝑧) = 𝑦
24		ssiun2 4499	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐴 → ({𝑥} × 𝐵) ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))
25	24	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ({𝑥} × 𝐵) ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))
26		simpr 476	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
27		vsnid 4156	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ {𝑥}
28		opelxp 5070	. . . . . . . . . . . . 13 ⊢ (⟨𝑥, 𝑦⟩ ∈ ({𝑥} × 𝐵) ↔ (𝑥 ∈ {𝑥} ∧ 𝑦 ∈ 𝐵))
29	27, 28	mpbiran 955	. . . . . . . . . . . 12 ⊢ (⟨𝑥, 𝑦⟩ ∈ ({𝑥} × 𝐵) ↔ 𝑦 ∈ 𝐵)
30	26, 29	sylibr 223	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ⟨𝑥, 𝑦⟩ ∈ ({𝑥} × 𝐵))
31	25, 30	sseldd 3569	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ⟨𝑥, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))
32	31, 9	syl6eleqr 2699	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ⟨𝑥, 𝑦⟩ ∈ 𝑇)
33		vex 3176	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
34		vex 3176	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
35	33, 34	op2nd 7068	. . . . . . . . 9 ⊢ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
36		fveq2 6103	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑧) = (2nd ‘⟨𝑥, 𝑦⟩))
37	36	eqeq1d 2612	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((2nd ‘𝑧) = 𝑦 ↔ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦))
38	37	rspcev 3282	. . . . . . . . 9 ⊢ ((⟨𝑥, 𝑦⟩ ∈ 𝑇 ∧ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦) → ∃𝑧 ∈ 𝑇 (2nd ‘𝑧) = 𝑦)
39	32, 35, 38	sylancl 693	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∃𝑧 ∈ 𝑇 (2nd ‘𝑧) = 𝑦)
40	39	ex 449	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → ∃𝑧 ∈ 𝑇 (2nd ‘𝑧) = 𝑦))
41	23, 40	rexlimi 3006	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → ∃𝑧 ∈ 𝑇 (2nd ‘𝑧) = 𝑦)
42	19, 41	impbii 198	. . . . 5 ⊢ (∃𝑧 ∈ 𝑇 (2nd ‘𝑧) = 𝑦 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
43		fvelimab 6163	. . . . . 6 ⊢ ((2nd Fn V ∧ 𝑇 ⊆ V) → (𝑦 ∈ (2nd “ 𝑇) ↔ ∃𝑧 ∈ 𝑇 (2nd ‘𝑧) = 𝑦))
44	4, 5, 43	mp2an 704	. . . . 5 ⊢ (𝑦 ∈ (2nd “ 𝑇) ↔ ∃𝑧 ∈ 𝑇 (2nd ‘𝑧) = 𝑦)
45		eliun 4460	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
46	42, 44, 45	3bitr4i 291	. . . 4 ⊢ (𝑦 ∈ (2nd “ 𝑇) ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
47	46	eqriv 2607	. . 3 ⊢ (2nd “ 𝑇) = ∪ 𝑥 ∈ 𝐴 𝐵
48	8, 47	eqtr3i 2634	. 2 ⊢ ran (2nd ↾ 𝑇) = ∪ 𝑥 ∈ 𝐴 𝐵
49		df-fo 5810	. 2 ⊢ ((2nd ↾ 𝑇):𝑇–onto→∪ 𝑥 ∈ 𝐴 𝐵 ↔ ((2nd ↾ 𝑇) Fn 𝑇 ∧ ran (2nd ↾ 𝑇) = ∪ 𝑥 ∈ 𝐴 𝐵))
50	7, 48, 49	mpbir2an 957	1 ⊢ (2nd ↾ 𝑇):𝑇–onto→∪ 𝑥 ∈ 𝐴 𝐵

Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  {csn 4125  ⟨cop 4131  ∪ ciun 4455   × cxp 5036  ran crn 5039   ↾ cres 5040   “ cima 5041   Fn wfn 5799  ⟶wf 5800  –onto→wfo 5802  ‘cfv 5804  2^nd c2nd 7058

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-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-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  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-fo 5810  df-fv 5812  df-2nd 7060

This theorem is referenced by:  iundomg  9242