Theorem iunfo 10492

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 7989	. . . 4 ⊢ 2nd :V–onto→V
2		fof 6772	. . . 4 ⊢ (2nd :V–onto→V → 2nd :V⟶V)
3		ffn 6688	. . . 4 ⊢ (2nd :V⟶V → 2nd Fn V)
4	1, 2, 3	mp2b 10	. . 3 ⊢ 2nd Fn V
5		ssv 3971	. . 3 ⊢ 𝑇 ⊆ V
6		fnssres 6641	. . 3 ⊢ ((2nd Fn V ∧ 𝑇 ⊆ V) → (2nd ↾ 𝑇) Fn 𝑇)
7	4, 5, 6	mp2an 692	. 2 ⊢ (2nd ↾ 𝑇) Fn 𝑇
8		df-ima 5651	. . 3 ⊢ (2nd “ 𝑇) = ran (2nd ↾ 𝑇)
9		iunfo.1	. . . . . . . . . . 11 ⊢ 𝑇 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
10	9	eleq2i 2820	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑇 ↔ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))
11		eliun 4959	. . . . . . . . . 10 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ ({𝑥} × 𝐵))
12	10, 11	bitri 275	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑇 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ ({𝑥} × 𝐵))
13		xp2nd 8001	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ({𝑥} × 𝐵) → (2nd ‘𝑧) ∈ 𝐵)
14		eleq1 2816	. . . . . . . . . . 11 ⊢ ((2nd ‘𝑧) = 𝑦 → ((2nd ‘𝑧) ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
15	13, 14	imbitrid 244	. . . . . . . . . 10 ⊢ ((2nd ‘𝑧) = 𝑦 → (𝑧 ∈ ({𝑥} × 𝐵) → 𝑦 ∈ 𝐵))
16	15	reximdv 3148	. . . . . . . . 9 ⊢ ((2nd ‘𝑧) = 𝑦 → (∃𝑥 ∈ 𝐴 𝑧 ∈ ({𝑥} × 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵))
17	12, 16	biimtrid 242	. . . . . . . 8 ⊢ ((2nd ‘𝑧) = 𝑦 → (𝑧 ∈ 𝑇 → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵))
18	17	impcom 407	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑇 ∧ (2nd ‘𝑧) = 𝑦) → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
19	18	rexlimiva 3126	. . . . . 6 ⊢ (∃𝑧 ∈ 𝑇 (2nd ‘𝑧) = 𝑦 → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
20		nfiu1 4991	. . . . . . . . 9 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
21	9, 20	nfcxfr 2889	. . . . . . . 8 ⊢ Ⅎ𝑥𝑇
22		nfv 1914	. . . . . . . 8 ⊢ Ⅎ𝑥(2nd ‘𝑧) = 𝑦
23	21, 22	nfrexw 3287	. . . . . . 7 ⊢ Ⅎ𝑥∃𝑧 ∈ 𝑇 (2nd ‘𝑧) = 𝑦
24		ssiun2 5011	. . . . . . . . . . . 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 484	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
27		vsnid 4627	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ {𝑥}
28		opelxp 5674	. . . . . . . . . . . . 13 ⊢ (⟨𝑥, 𝑦⟩ ∈ ({𝑥} × 𝐵) ↔ (𝑥 ∈ {𝑥} ∧ 𝑦 ∈ 𝐵))
29	27, 28	mpbiran 709	. . . . . . . . . . . 12 ⊢ (⟨𝑥, 𝑦⟩ ∈ ({𝑥} × 𝐵) ↔ 𝑦 ∈ 𝐵)
30	26, 29	sylibr 234	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ⟨𝑥, 𝑦⟩ ∈ ({𝑥} × 𝐵))
31	25, 30	sseldd 3947	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ⟨𝑥, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))
32	31, 9	eleqtrrdi 2839	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ⟨𝑥, 𝑦⟩ ∈ 𝑇)
33		vex 3451	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
34		vex 3451	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
35	33, 34	op2nd 7977	. . . . . . . . 9 ⊢ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
36		fveqeq2 6867	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((2nd ‘𝑧) = 𝑦 ↔ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦))
37	36	rspcev 3588	. . . . . . . . 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 412	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → ∃𝑧 ∈ 𝑇 (2nd ‘𝑧) = 𝑦))
40	23, 39	rexlimi 3237	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → ∃𝑧 ∈ 𝑇 (2nd ‘𝑧) = 𝑦)
41	19, 40	impbii 209	. . . . 5 ⊢ (∃𝑧 ∈ 𝑇 (2nd ‘𝑧) = 𝑦 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
42		fvelimab 6933	. . . . . 6 ⊢ ((2nd Fn V ∧ 𝑇 ⊆ V) → (𝑦 ∈ (2nd “ 𝑇) ↔ ∃𝑧 ∈ 𝑇 (2nd ‘𝑧) = 𝑦))
43	4, 5, 42	mp2an 692	. . . . 5 ⊢ (𝑦 ∈ (2nd “ 𝑇) ↔ ∃𝑧 ∈ 𝑇 (2nd ‘𝑧) = 𝑦)
44		eliun 4959	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
45	41, 43, 44	3bitr4i 303	. . . 4 ⊢ (𝑦 ∈ (2nd “ 𝑇) ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
46	45	eqriv 2726	. . 3 ⊢ (2nd “ 𝑇) = ∪ 𝑥 ∈ 𝐴 𝐵
47	8, 46	eqtr3i 2754	. 2 ⊢ ran (2nd ↾ 𝑇) = ∪ 𝑥 ∈ 𝐴 𝐵
48		df-fo 6517	. 2 ⊢ ((2nd ↾ 𝑇):𝑇–onto→∪ 𝑥 ∈ 𝐴 𝐵 ↔ ((2nd ↾ 𝑇) Fn 𝑇 ∧ ran (2nd ↾ 𝑇) = ∪ 𝑥 ∈ 𝐴 𝐵))
49	7, 47, 48	mpbir2an 711	1 ⊢ (2nd ↾ 𝑇):𝑇–onto→∪ 𝑥 ∈ 𝐴 𝐵

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  Vcvv 3447   ⊆ wss 3914  {csn 4589  ⟨cop 4595  ∪ ciun 4955   × cxp 5636  ran crn 5639   ↾ cres 5640   “ cima 5641   Fn wfn 6506  ⟶wf 6507  –onto→wfo 6509  ‘cfv 6511  2^nd c2nd 7967

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 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

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 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fo 6517  df-fv 6519  df-2nd 7969

This theorem is referenced by:  iundomg  10494  2ndresdjuf1o  32574