Theorem iunfo 10518

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 7980	. . . 4 ⊢ 2nd :V–onto→V
2		fof 6793	. . . 4 ⊢ (2nd :V–onto→V → 2nd :V⟶V)
3		ffn 6705	. . . 4 ⊢ (2nd :V⟶V → 2nd Fn V)
4	1, 2, 3	mp2b 10	. . 3 ⊢ 2nd Fn V
5		ssv 4003	. . 3 ⊢ 𝑇 ⊆ V
6		fnssres 6661	. . 3 ⊢ ((2nd Fn V ∧ 𝑇 ⊆ V) → (2nd ↾ 𝑇) Fn 𝑇)
7	4, 5, 6	mp2an 690	. 2 ⊢ (2nd ↾ 𝑇) Fn 𝑇
8		df-ima 5683	. . 3 ⊢ (2nd “ 𝑇) = ran (2nd ↾ 𝑇)
9		iunfo.1	. . . . . . . . . . 11 ⊢ 𝑇 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
10	9	eleq2i 2825	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑇 ↔ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))
11		eliun 4995	. . . . . . . . . 10 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ ({𝑥} × 𝐵))
12	10, 11	bitri 274	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑇 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ ({𝑥} × 𝐵))
13		xp2nd 7992	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ({𝑥} × 𝐵) → (2nd ‘𝑧) ∈ 𝐵)
14		eleq1 2821	. . . . . . . . . . 11 ⊢ ((2nd ‘𝑧) = 𝑦 → ((2nd ‘𝑧) ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
15	13, 14	imbitrid 243	. . . . . . . . . 10 ⊢ ((2nd ‘𝑧) = 𝑦 → (𝑧 ∈ ({𝑥} × 𝐵) → 𝑦 ∈ 𝐵))
16	15	reximdv 3170	. . . . . . . . 9 ⊢ ((2nd ‘𝑧) = 𝑦 → (∃𝑥 ∈ 𝐴 𝑧 ∈ ({𝑥} × 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵))
17	12, 16	biimtrid 241	. . . . . . . 8 ⊢ ((2nd ‘𝑧) = 𝑦 → (𝑧 ∈ 𝑇 → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵))
18	17	impcom 408	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑇 ∧ (2nd ‘𝑧) = 𝑦) → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
19	18	rexlimiva 3147	. . . . . 6 ⊢ (∃𝑧 ∈ 𝑇 (2nd ‘𝑧) = 𝑦 → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
20		nfiu1 5025	. . . . . . . . 9 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
21	9, 20	nfcxfr 2901	. . . . . . . 8 ⊢ Ⅎ𝑥𝑇
22		nfv 1917	. . . . . . . 8 ⊢ Ⅎ𝑥(2nd ‘𝑧) = 𝑦
23	21, 22	nfrexw 3310	. . . . . . 7 ⊢ Ⅎ𝑥∃𝑧 ∈ 𝑇 (2nd ‘𝑧) = 𝑦
24		ssiun2 5044	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐴 → ({𝑥} × 𝐵) ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))
25	24	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ({𝑥} × 𝐵) ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))
26		simpr 485	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
27		vsnid 4660	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ {𝑥}
28		opelxp 5706	. . . . . . . . . . . . 13 ⊢ (⟨𝑥, 𝑦⟩ ∈ ({𝑥} × 𝐵) ↔ (𝑥 ∈ {𝑥} ∧ 𝑦 ∈ 𝐵))
29	27, 28	mpbiran 707	. . . . . . . . . . . 12 ⊢ (⟨𝑥, 𝑦⟩ ∈ ({𝑥} × 𝐵) ↔ 𝑦 ∈ 𝐵)
30	26, 29	sylibr 233	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ⟨𝑥, 𝑦⟩ ∈ ({𝑥} × 𝐵))
31	25, 30	sseldd 3980	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ⟨𝑥, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))
32	31, 9	eleqtrrdi 2844	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ⟨𝑥, 𝑦⟩ ∈ 𝑇)
33		vex 3478	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
34		vex 3478	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
35	33, 34	op2nd 7968	. . . . . . . . 9 ⊢ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
36		fveqeq2 6888	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((2nd ‘𝑧) = 𝑦 ↔ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦))
37	36	rspcev 3610	. . . . . . . . 9 ⊢ ((⟨𝑥, 𝑦⟩ ∈ 𝑇 ∧ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦) → ∃𝑧 ∈ 𝑇 (2nd ‘𝑧) = 𝑦)
38	32, 35, 37	sylancl 586	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∃𝑧 ∈ 𝑇 (2nd ‘𝑧) = 𝑦)
39	38	ex 413	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → ∃𝑧 ∈ 𝑇 (2nd ‘𝑧) = 𝑦))
40	23, 39	rexlimi 3256	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → ∃𝑧 ∈ 𝑇 (2nd ‘𝑧) = 𝑦)
41	19, 40	impbii 208	. . . . 5 ⊢ (∃𝑧 ∈ 𝑇 (2nd ‘𝑧) = 𝑦 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
42		fvelimab 6951	. . . . . 6 ⊢ ((2nd Fn V ∧ 𝑇 ⊆ V) → (𝑦 ∈ (2nd “ 𝑇) ↔ ∃𝑧 ∈ 𝑇 (2nd ‘𝑧) = 𝑦))
43	4, 5, 42	mp2an 690	. . . . 5 ⊢ (𝑦 ∈ (2nd “ 𝑇) ↔ ∃𝑧 ∈ 𝑇 (2nd ‘𝑧) = 𝑦)
44		eliun 4995	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
45	41, 43, 44	3bitr4i 302	. . . 4 ⊢ (𝑦 ∈ (2nd “ 𝑇) ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
46	45	eqriv 2729	. . 3 ⊢ (2nd “ 𝑇) = ∪ 𝑥 ∈ 𝐴 𝐵
47	8, 46	eqtr3i 2762	. 2 ⊢ ran (2nd ↾ 𝑇) = ∪ 𝑥 ∈ 𝐴 𝐵
48		df-fo 6539	. 2 ⊢ ((2nd ↾ 𝑇):𝑇–onto→∪ 𝑥 ∈ 𝐴 𝐵 ↔ ((2nd ↾ 𝑇) Fn 𝑇 ∧ ran (2nd ↾ 𝑇) = ∪ 𝑥 ∈ 𝐴 𝐵))
49	7, 47, 48	mpbir2an 709	1 ⊢ (2nd ↾ 𝑇):𝑇–onto→∪ 𝑥 ∈ 𝐴 𝐵

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∃wrex 3070  Vcvv 3474   ⊆ wss 3945  {csn 4623  ⟨cop 4629  ∪ ciun 4991   × cxp 5668  ran crn 5671   ↾ cres 5672   “ cima 5673   Fn wfn 6528  ⟶wf 6529  –onto→wfo 6531  ‘cfv 6533  2^nd c2nd 7958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5293  ax-nul 5300  ax-pr 5421  ax-un 7709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-fo 6539  df-fv 6541  df-2nd 7960

This theorem is referenced by:  iundomg  10520  2ndresdjuf1o  31808