MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iunfo Structured version   Visualization version   GIF version

Theorem iunfo 10295
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 = 𝑥𝐴 ({𝑥} × 𝐵)
Assertion
Ref Expression
iunfo (2nd𝑇):𝑇onto 𝑥𝐴 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.
StepHypRef Expression
1 fo2nd 7852 . . . 4 2nd :V–onto→V
2 fof 6688 . . . 4 (2nd :V–onto→V → 2nd :V⟶V)
3 ffn 6600 . . . 4 (2nd :V⟶V → 2nd Fn V)
41, 2, 3mp2b 10 . . 3 2nd Fn V
5 ssv 3945 . . 3 𝑇 ⊆ V
6 fnssres 6555 . . 3 ((2nd Fn V ∧ 𝑇 ⊆ V) → (2nd𝑇) Fn 𝑇)
74, 5, 6mp2an 689 . 2 (2nd𝑇) Fn 𝑇
8 df-ima 5602 . . 3 (2nd𝑇) = ran (2nd𝑇)
9 iunfo.1 . . . . . . . . . . 11 𝑇 = 𝑥𝐴 ({𝑥} × 𝐵)
109eleq2i 2830 . . . . . . . . . 10 (𝑧𝑇𝑧 𝑥𝐴 ({𝑥} × 𝐵))
11 eliun 4928 . . . . . . . . . 10 (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↔ ∃𝑥𝐴 𝑧 ∈ ({𝑥} × 𝐵))
1210, 11bitri 274 . . . . . . . . 9 (𝑧𝑇 ↔ ∃𝑥𝐴 𝑧 ∈ ({𝑥} × 𝐵))
13 xp2nd 7864 . . . . . . . . . . 11 (𝑧 ∈ ({𝑥} × 𝐵) → (2nd𝑧) ∈ 𝐵)
14 eleq1 2826 . . . . . . . . . . 11 ((2nd𝑧) = 𝑦 → ((2nd𝑧) ∈ 𝐵𝑦𝐵))
1513, 14syl5ib 243 . . . . . . . . . 10 ((2nd𝑧) = 𝑦 → (𝑧 ∈ ({𝑥} × 𝐵) → 𝑦𝐵))
1615reximdv 3202 . . . . . . . . 9 ((2nd𝑧) = 𝑦 → (∃𝑥𝐴 𝑧 ∈ ({𝑥} × 𝐵) → ∃𝑥𝐴 𝑦𝐵))
1712, 16syl5bi 241 . . . . . . . 8 ((2nd𝑧) = 𝑦 → (𝑧𝑇 → ∃𝑥𝐴 𝑦𝐵))
1817impcom 408 . . . . . . 7 ((𝑧𝑇 ∧ (2nd𝑧) = 𝑦) → ∃𝑥𝐴 𝑦𝐵)
1918rexlimiva 3210 . . . . . 6 (∃𝑧𝑇 (2nd𝑧) = 𝑦 → ∃𝑥𝐴 𝑦𝐵)
20 nfiu1 4958 . . . . . . . . 9 𝑥 𝑥𝐴 ({𝑥} × 𝐵)
219, 20nfcxfr 2905 . . . . . . . 8 𝑥𝑇
22 nfv 1917 . . . . . . . 8 𝑥(2nd𝑧) = 𝑦
2321, 22nfrex 3242 . . . . . . 7 𝑥𝑧𝑇 (2nd𝑧) = 𝑦
24 ssiun2 4977 . . . . . . . . . . . 12 (𝑥𝐴 → ({𝑥} × 𝐵) ⊆ 𝑥𝐴 ({𝑥} × 𝐵))
2524adantr 481 . . . . . . . . . . 11 ((𝑥𝐴𝑦𝐵) → ({𝑥} × 𝐵) ⊆ 𝑥𝐴 ({𝑥} × 𝐵))
26 simpr 485 . . . . . . . . . . . 12 ((𝑥𝐴𝑦𝐵) → 𝑦𝐵)
27 vsnid 4598 . . . . . . . . . . . . 13 𝑥 ∈ {𝑥}
28 opelxp 5625 . . . . . . . . . . . . 13 (⟨𝑥, 𝑦⟩ ∈ ({𝑥} × 𝐵) ↔ (𝑥 ∈ {𝑥} ∧ 𝑦𝐵))
2927, 28mpbiran 706 . . . . . . . . . . . 12 (⟨𝑥, 𝑦⟩ ∈ ({𝑥} × 𝐵) ↔ 𝑦𝐵)
3026, 29sylibr 233 . . . . . . . . . . 11 ((𝑥𝐴𝑦𝐵) → ⟨𝑥, 𝑦⟩ ∈ ({𝑥} × 𝐵))
3125, 30sseldd 3922 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐵) → ⟨𝑥, 𝑦⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵))
3231, 9eleqtrrdi 2850 . . . . . . . . 9 ((𝑥𝐴𝑦𝐵) → ⟨𝑥, 𝑦⟩ ∈ 𝑇)
33 vex 3436 . . . . . . . . . 10 𝑥 ∈ V
34 vex 3436 . . . . . . . . . 10 𝑦 ∈ V
3533, 34op2nd 7840 . . . . . . . . 9 (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
36 fveqeq2 6783 . . . . . . . . . 10 (𝑧 = ⟨𝑥, 𝑦⟩ → ((2nd𝑧) = 𝑦 ↔ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦))
3736rspcev 3561 . . . . . . . . 9 ((⟨𝑥, 𝑦⟩ ∈ 𝑇 ∧ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦) → ∃𝑧𝑇 (2nd𝑧) = 𝑦)
3832, 35, 37sylancl 586 . . . . . . . 8 ((𝑥𝐴𝑦𝐵) → ∃𝑧𝑇 (2nd𝑧) = 𝑦)
3938ex 413 . . . . . . 7 (𝑥𝐴 → (𝑦𝐵 → ∃𝑧𝑇 (2nd𝑧) = 𝑦))
4023, 39rexlimi 3248 . . . . . 6 (∃𝑥𝐴 𝑦𝐵 → ∃𝑧𝑇 (2nd𝑧) = 𝑦)
4119, 40impbii 208 . . . . 5 (∃𝑧𝑇 (2nd𝑧) = 𝑦 ↔ ∃𝑥𝐴 𝑦𝐵)
42 fvelimab 6841 . . . . . 6 ((2nd Fn V ∧ 𝑇 ⊆ V) → (𝑦 ∈ (2nd𝑇) ↔ ∃𝑧𝑇 (2nd𝑧) = 𝑦))
434, 5, 42mp2an 689 . . . . 5 (𝑦 ∈ (2nd𝑇) ↔ ∃𝑧𝑇 (2nd𝑧) = 𝑦)
44 eliun 4928 . . . . 5 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
4541, 43, 443bitr4i 303 . . . 4 (𝑦 ∈ (2nd𝑇) ↔ 𝑦 𝑥𝐴 𝐵)
4645eqriv 2735 . . 3 (2nd𝑇) = 𝑥𝐴 𝐵
478, 46eqtr3i 2768 . 2 ran (2nd𝑇) = 𝑥𝐴 𝐵
48 df-fo 6439 . 2 ((2nd𝑇):𝑇onto 𝑥𝐴 𝐵 ↔ ((2nd𝑇) Fn 𝑇 ∧ ran (2nd𝑇) = 𝑥𝐴 𝐵))
497, 47, 48mpbir2an 708 1 (2nd𝑇):𝑇onto 𝑥𝐴 𝐵
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1539  wcel 2106  wrex 3065  Vcvv 3432  wss 3887  {csn 4561  cop 4567   ciun 4924   × cxp 5587  ran crn 5590  cres 5591  cima 5592   Fn wfn 6428  wf 6429  ontowfo 6431  cfv 6433  2nd c2nd 7830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fo 6439  df-fv 6441  df-2nd 7832
This theorem is referenced by:  iundomg  10297  2ndresdjuf1o  30987
  Copyright terms: Public domain W3C validator