Theorem iunfo 10548

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 8008	. . . 4 ⊢ 2nd :V–onto→V
2		fof 6790	. . . 4 ⊢ (2nd :V–onto→V → 2nd :V⟶V)
3		ffn 6703	. . . 4 ⊢ (2nd :V⟶V → 2nd Fn V)
4	1, 2, 3	mp2b 10	. . 3 ⊢ 2nd Fn V
5		ssv 3955	. . 3 ⊢ 𝑇 ⊆ V
6		fnssres 6656	. . 3 ⊢ ((2nd Fn V ∧ 𝑇 ⊆ V) → (2nd ↾ 𝑇) Fn 𝑇)
7	4, 5, 6	mp2an 705	. 2 ⊢ (2nd ↾ 𝑇) Fn 𝑇
8		df-ima 5668	. . 3 ⊢ (2nd “ 𝑇) = ran (2nd ↾ 𝑇)
9		iunfo.1	. . . . . . . . . . 11 ⊢ 𝑇 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
10	9	eleq2i 2852	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑇 ↔ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))
11		eliun 4955	. . . . . . . . . 10 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ ({𝑥} × 𝐵))
12	10, 11	bitri 278	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑇 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ ({𝑥} × 𝐵))
13		xp2nd 8020	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ({𝑥} × 𝐵) → (2nd ‘𝑧) ∈ 𝐵)
14		eleq1 2848	. . . . . . . . . . 11 ⊢ ((2nd ‘𝑧) = 𝑦 → ((2nd ‘𝑧) ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
15	13, 14	imbitrid 247	. . . . . . . . . 10 ⊢ ((2nd ‘𝑧) = 𝑦 → (𝑧 ∈ ({𝑥} × 𝐵) → 𝑦 ∈ 𝐵))
16	15	reximdv 3177	. . . . . . . . 9 ⊢ ((2nd ‘𝑧) = 𝑦 → (∃𝑥 ∈ 𝐴 𝑧 ∈ ({𝑥} × 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵))
17	12, 16	biimtrid 245	. . . . . . . 8 ⊢ ((2nd ‘𝑧) = 𝑦 → (𝑧 ∈ 𝑇 → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵))
18	17	impcom 413	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑇 ∧ (2nd ‘𝑧) = 𝑦) → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
19	18	rexlimiva 3155	. . . . . 6 ⊢ (∃𝑧 ∈ 𝑇 (2nd ‘𝑧) = 𝑦 → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
20		nfiu1 4986	. . . . . . . . 9 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
21	9, 20	nfcxfr 2920	. . . . . . . 8 ⊢ Ⅎ𝑥𝑇
22		nfv 1947	. . . . . . . 8 ⊢ Ⅎ𝑥(2nd ‘𝑧) = 𝑦
23	21, 22	nfrexw 3310	. . . . . . 7 ⊢ Ⅎ𝑥∃𝑧 ∈ 𝑇 (2nd ‘𝑧) = 𝑦
24		ssiun2 5006	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐴 → ({𝑥} × 𝐵) ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))
25	24	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ({𝑥} × 𝐵) ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))
26		vsnid 4624	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ {𝑥}
27		opelxp 5691	. . . . . . . . . . . . 13 ⊢ (⟨𝑥, 𝑦⟩ ∈ ({𝑥} × 𝐵) ↔ (𝑥 ∈ {𝑥} ∧ 𝑦 ∈ 𝐵))
28	26, 27	mpbiran 722	. . . . . . . . . . . 12 ⊢ (⟨𝑥, 𝑦⟩ ∈ ({𝑥} × 𝐵) ↔ 𝑦 ∈ 𝐵)
29	28	bilanri 512	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ⟨𝑥, 𝑦⟩ ∈ ({𝑥} × 𝐵))
30	25, 29	sseldd 3932	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ⟨𝑥, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))
31	30, 9	eleqtrrdi 2871	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ⟨𝑥, 𝑦⟩ ∈ 𝑇)
32		vex 3454	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
33		vex 3454	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
34	32, 33	op2nd 7996	. . . . . . . . 9 ⊢ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
35		fveqeq2 6888	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((2nd ‘𝑧) = 𝑦 ↔ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦))
36	35	rspcev 3576	. . . . . . . . 9 ⊢ ((⟨𝑥, 𝑦⟩ ∈ 𝑇 ∧ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦) → ∃𝑧 ∈ 𝑇 (2nd ‘𝑧) = 𝑦)
37	31, 34, 36	sylancl 598	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∃𝑧 ∈ 𝑇 (2nd ‘𝑧) = 𝑦)
38	37	ex 418	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → ∃𝑧 ∈ 𝑇 (2nd ‘𝑧) = 𝑦))
39	23, 38	rexlimi 3262	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → ∃𝑧 ∈ 𝑇 (2nd ‘𝑧) = 𝑦)
40	19, 39	impbii 212	. . . . 5 ⊢ (∃𝑧 ∈ 𝑇 (2nd ‘𝑧) = 𝑦 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
41		fvelimab 6951	. . . . . 6 ⊢ ((2nd Fn V ∧ 𝑇 ⊆ V) → (𝑦 ∈ (2nd “ 𝑇) ↔ ∃𝑧 ∈ 𝑇 (2nd ‘𝑧) = 𝑦))
42	4, 5, 41	mp2an 705	. . . . 5 ⊢ (𝑦 ∈ (2nd “ 𝑇) ↔ ∃𝑧 ∈ 𝑇 (2nd ‘𝑧) = 𝑦)
43		eliun 4955	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
44	40, 42, 43	3bitr4i 306	. . . 4 ⊢ (𝑦 ∈ (2nd “ 𝑇) ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
45	44	eqriv 2757	. . 3 ⊢ (2nd “ 𝑇) = ∪ 𝑥 ∈ 𝐴 𝐵
46	8, 45	eqtr3i 2785	. 2 ⊢ ran (2nd ↾ 𝑇) = ∪ 𝑥 ∈ 𝐴 𝐵
47		df-fo 6539	. 2 ⊢ ((2nd ↾ 𝑇):𝑇–onto→∪ 𝑥 ∈ 𝐴 𝐵 ↔ ((2nd ↾ 𝑇) Fn 𝑇 ∧ ran (2nd ↾ 𝑇) = ∪ 𝑥 ∈ 𝐴 𝐵))
48	7, 46, 47	mpbir2an 724	1 ⊢ (2nd ↾ 𝑇):𝑇–onto→∪ 𝑥 ∈ 𝐴 𝐵

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3086  Vcvv 3450   ⊆ wss 3899  {csn 4584  ⟨cop 4590  ∪ ciun 4951   × cxp 5653  ran crn 5656   ↾ cres 5657   “ cima 5658   Fn wfn 6528  ⟶wf 6529  –onto→wfo 6531  ‘cfv 6533  2^nd c2nd 7986

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fo 6539  df-fv 6541  df-2nd 7988

This theorem is used by:  iundomg  10550  2ndresdjuf1o  33124