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

Theorem iunfo 10576
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 8033 . . . 4 2nd :V–onto→V
2 fof 6820 . . . 4 (2nd :V–onto→V → 2nd :V⟶V)
3 ffn 6736 . . . 4 (2nd :V⟶V → 2nd Fn V)
41, 2, 3mp2b 10 . . 3 2nd Fn V
5 ssv 4019 . . 3 𝑇 ⊆ V
6 fnssres 6691 . . 3 ((2nd Fn V ∧ 𝑇 ⊆ V) → (2nd𝑇) Fn 𝑇)
74, 5, 6mp2an 692 . 2 (2nd𝑇) Fn 𝑇
8 df-ima 5701 . . 3 (2nd𝑇) = ran (2nd𝑇)
9 iunfo.1 . . . . . . . . . . 11 𝑇 = 𝑥𝐴 ({𝑥} × 𝐵)
109eleq2i 2830 . . . . . . . . . 10 (𝑧𝑇𝑧 𝑥𝐴 ({𝑥} × 𝐵))
11 eliun 4999 . . . . . . . . . 10 (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↔ ∃𝑥𝐴 𝑧 ∈ ({𝑥} × 𝐵))
1210, 11bitri 275 . . . . . . . . 9 (𝑧𝑇 ↔ ∃𝑥𝐴 𝑧 ∈ ({𝑥} × 𝐵))
13 xp2nd 8045 . . . . . . . . . . 11 (𝑧 ∈ ({𝑥} × 𝐵) → (2nd𝑧) ∈ 𝐵)
14 eleq1 2826 . . . . . . . . . . 11 ((2nd𝑧) = 𝑦 → ((2nd𝑧) ∈ 𝐵𝑦𝐵))
1513, 14imbitrid 244 . . . . . . . . . 10 ((2nd𝑧) = 𝑦 → (𝑧 ∈ ({𝑥} × 𝐵) → 𝑦𝐵))
1615reximdv 3167 . . . . . . . . 9 ((2nd𝑧) = 𝑦 → (∃𝑥𝐴 𝑧 ∈ ({𝑥} × 𝐵) → ∃𝑥𝐴 𝑦𝐵))
1712, 16biimtrid 242 . . . . . . . 8 ((2nd𝑧) = 𝑦 → (𝑧𝑇 → ∃𝑥𝐴 𝑦𝐵))
1817impcom 407 . . . . . . 7 ((𝑧𝑇 ∧ (2nd𝑧) = 𝑦) → ∃𝑥𝐴 𝑦𝐵)
1918rexlimiva 3144 . . . . . 6 (∃𝑧𝑇 (2nd𝑧) = 𝑦 → ∃𝑥𝐴 𝑦𝐵)
20 nfiu1 5031 . . . . . . . . 9 𝑥 𝑥𝐴 ({𝑥} × 𝐵)
219, 20nfcxfr 2900 . . . . . . . 8 𝑥𝑇
22 nfv 1911 . . . . . . . 8 𝑥(2nd𝑧) = 𝑦
2321, 22nfrexw 3310 . . . . . . 7 𝑥𝑧𝑇 (2nd𝑧) = 𝑦
24 ssiun2 5051 . . . . . . . . . . . 12 (𝑥𝐴 → ({𝑥} × 𝐵) ⊆ 𝑥𝐴 ({𝑥} × 𝐵))
2524adantr 480 . . . . . . . . . . 11 ((𝑥𝐴𝑦𝐵) → ({𝑥} × 𝐵) ⊆ 𝑥𝐴 ({𝑥} × 𝐵))
26 simpr 484 . . . . . . . . . . . 12 ((𝑥𝐴𝑦𝐵) → 𝑦𝐵)
27 vsnid 4667 . . . . . . . . . . . . 13 𝑥 ∈ {𝑥}
28 opelxp 5724 . . . . . . . . . . . . 13 (⟨𝑥, 𝑦⟩ ∈ ({𝑥} × 𝐵) ↔ (𝑥 ∈ {𝑥} ∧ 𝑦𝐵))
2927, 28mpbiran 709 . . . . . . . . . . . 12 (⟨𝑥, 𝑦⟩ ∈ ({𝑥} × 𝐵) ↔ 𝑦𝐵)
3026, 29sylibr 234 . . . . . . . . . . 11 ((𝑥𝐴𝑦𝐵) → ⟨𝑥, 𝑦⟩ ∈ ({𝑥} × 𝐵))
3125, 30sseldd 3995 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐵) → ⟨𝑥, 𝑦⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵))
3231, 9eleqtrrdi 2849 . . . . . . . . 9 ((𝑥𝐴𝑦𝐵) → ⟨𝑥, 𝑦⟩ ∈ 𝑇)
33 vex 3481 . . . . . . . . . 10 𝑥 ∈ V
34 vex 3481 . . . . . . . . . 10 𝑦 ∈ V
3533, 34op2nd 8021 . . . . . . . . 9 (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
36 fveqeq2 6915 . . . . . . . . . 10 (𝑧 = ⟨𝑥, 𝑦⟩ → ((2nd𝑧) = 𝑦 ↔ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦))
3736rspcev 3621 . . . . . . . . 9 ((⟨𝑥, 𝑦⟩ ∈ 𝑇 ∧ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦) → ∃𝑧𝑇 (2nd𝑧) = 𝑦)
3832, 35, 37sylancl 586 . . . . . . . 8 ((𝑥𝐴𝑦𝐵) → ∃𝑧𝑇 (2nd𝑧) = 𝑦)
3938ex 412 . . . . . . 7 (𝑥𝐴 → (𝑦𝐵 → ∃𝑧𝑇 (2nd𝑧) = 𝑦))
4023, 39rexlimi 3256 . . . . . 6 (∃𝑥𝐴 𝑦𝐵 → ∃𝑧𝑇 (2nd𝑧) = 𝑦)
4119, 40impbii 209 . . . . 5 (∃𝑧𝑇 (2nd𝑧) = 𝑦 ↔ ∃𝑥𝐴 𝑦𝐵)
42 fvelimab 6980 . . . . . 6 ((2nd Fn V ∧ 𝑇 ⊆ V) → (𝑦 ∈ (2nd𝑇) ↔ ∃𝑧𝑇 (2nd𝑧) = 𝑦))
434, 5, 42mp2an 692 . . . . 5 (𝑦 ∈ (2nd𝑇) ↔ ∃𝑧𝑇 (2nd𝑧) = 𝑦)
44 eliun 4999 . . . . 5 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
4541, 43, 443bitr4i 303 . . . 4 (𝑦 ∈ (2nd𝑇) ↔ 𝑦 𝑥𝐴 𝐵)
4645eqriv 2731 . . 3 (2nd𝑇) = 𝑥𝐴 𝐵
478, 46eqtr3i 2764 . 2 ran (2nd𝑇) = 𝑥𝐴 𝐵
48 df-fo 6568 . 2 ((2nd𝑇):𝑇onto 𝑥𝐴 𝐵 ↔ ((2nd𝑇) Fn 𝑇 ∧ ran (2nd𝑇) = 𝑥𝐴 𝐵))
497, 47, 48mpbir2an 711 1 (2nd𝑇):𝑇onto 𝑥𝐴 𝐵
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1536  wcel 2105  wrex 3067  Vcvv 3477  wss 3962  {csn 4630  cop 4636   ciun 4995   × cxp 5686  ran crn 5689  cres 5690  cima 5691   Fn wfn 6557  wf 6558  ontowfo 6560  cfv 6562  2nd c2nd 8011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fo 6568  df-fv 6570  df-2nd 8013
This theorem is referenced by:  iundomg  10578  2ndresdjuf1o  32666
  Copyright terms: Public domain W3C validator