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Theorem iunfo 9240
 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 7080 . . . 4 2nd :V–onto→V
2 fof 6028 . . . 4 (2nd :V–onto→V → 2nd :V⟶V)
3 ffn 5958 . . . 4 (2nd :V⟶V → 2nd Fn V)
41, 2, 3mp2b 10 . . 3 2nd Fn V
5 ssv 3588 . . 3 𝑇 ⊆ V
6 fnssres 5918 . . 3 ((2nd Fn V ∧ 𝑇 ⊆ V) → (2nd𝑇) Fn 𝑇)
74, 5, 6mp2an 704 . 2 (2nd𝑇) Fn 𝑇
8 df-ima 5051 . . 3 (2nd𝑇) = ran (2nd𝑇)
9 iunfo.1 . . . . . . . . . . 11 𝑇 = 𝑥𝐴 ({𝑥} × 𝐵)
109eleq2i 2680 . . . . . . . . . 10 (𝑧𝑇𝑧 𝑥𝐴 ({𝑥} × 𝐵))
11 eliun 4460 . . . . . . . . . 10 (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↔ ∃𝑥𝐴 𝑧 ∈ ({𝑥} × 𝐵))
1210, 11bitri 263 . . . . . . . . 9 (𝑧𝑇 ↔ ∃𝑥𝐴 𝑧 ∈ ({𝑥} × 𝐵))
13 xp2nd 7090 . . . . . . . . . . 11 (𝑧 ∈ ({𝑥} × 𝐵) → (2nd𝑧) ∈ 𝐵)
14 eleq1 2676 . . . . . . . . . . 11 ((2nd𝑧) = 𝑦 → ((2nd𝑧) ∈ 𝐵𝑦𝐵))
1513, 14syl5ib 233 . . . . . . . . . 10 ((2nd𝑧) = 𝑦 → (𝑧 ∈ ({𝑥} × 𝐵) → 𝑦𝐵))
1615reximdv 2999 . . . . . . . . 9 ((2nd𝑧) = 𝑦 → (∃𝑥𝐴 𝑧 ∈ ({𝑥} × 𝐵) → ∃𝑥𝐴 𝑦𝐵))
1712, 16syl5bi 231 . . . . . . . 8 ((2nd𝑧) = 𝑦 → (𝑧𝑇 → ∃𝑥𝐴 𝑦𝐵))
1817impcom 445 . . . . . . 7 ((𝑧𝑇 ∧ (2nd𝑧) = 𝑦) → ∃𝑥𝐴 𝑦𝐵)
1918rexlimiva 3010 . . . . . 6 (∃𝑧𝑇 (2nd𝑧) = 𝑦 → ∃𝑥𝐴 𝑦𝐵)
20 nfiu1 4486 . . . . . . . . 9 𝑥 𝑥𝐴 ({𝑥} × 𝐵)
219, 20nfcxfr 2749 . . . . . . . 8 𝑥𝑇
22 nfv 1830 . . . . . . . 8 𝑥(2nd𝑧) = 𝑦
2321, 22nfrex 2990 . . . . . . 7 𝑥𝑧𝑇 (2nd𝑧) = 𝑦
24 ssiun2 4499 . . . . . . . . . . . 12 (𝑥𝐴 → ({𝑥} × 𝐵) ⊆ 𝑥𝐴 ({𝑥} × 𝐵))
2524adantr 480 . . . . . . . . . . 11 ((𝑥𝐴𝑦𝐵) → ({𝑥} × 𝐵) ⊆ 𝑥𝐴 ({𝑥} × 𝐵))
26 simpr 476 . . . . . . . . . . . 12 ((𝑥𝐴𝑦𝐵) → 𝑦𝐵)
27 vsnid 4156 . . . . . . . . . . . . 13 𝑥 ∈ {𝑥}
28 opelxp 5070 . . . . . . . . . . . . 13 (⟨𝑥, 𝑦⟩ ∈ ({𝑥} × 𝐵) ↔ (𝑥 ∈ {𝑥} ∧ 𝑦𝐵))
2927, 28mpbiran 955 . . . . . . . . . . . 12 (⟨𝑥, 𝑦⟩ ∈ ({𝑥} × 𝐵) ↔ 𝑦𝐵)
3026, 29sylibr 223 . . . . . . . . . . 11 ((𝑥𝐴𝑦𝐵) → ⟨𝑥, 𝑦⟩ ∈ ({𝑥} × 𝐵))
3125, 30sseldd 3569 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐵) → ⟨𝑥, 𝑦⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵))
3231, 9syl6eleqr 2699 . . . . . . . . 9 ((𝑥𝐴𝑦𝐵) → ⟨𝑥, 𝑦⟩ ∈ 𝑇)
33 vex 3176 . . . . . . . . . 10 𝑥 ∈ V
34 vex 3176 . . . . . . . . . 10 𝑦 ∈ V
3533, 34op2nd 7068 . . . . . . . . 9 (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
36 fveq2 6103 . . . . . . . . . . 11 (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd𝑧) = (2nd ‘⟨𝑥, 𝑦⟩))
3736eqeq1d 2612 . . . . . . . . . 10 (𝑧 = ⟨𝑥, 𝑦⟩ → ((2nd𝑧) = 𝑦 ↔ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦))
3837rspcev 3282 . . . . . . . . 9 ((⟨𝑥, 𝑦⟩ ∈ 𝑇 ∧ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦) → ∃𝑧𝑇 (2nd𝑧) = 𝑦)
3932, 35, 38sylancl 693 . . . . . . . 8 ((𝑥𝐴𝑦𝐵) → ∃𝑧𝑇 (2nd𝑧) = 𝑦)
4039ex 449 . . . . . . 7 (𝑥𝐴 → (𝑦𝐵 → ∃𝑧𝑇 (2nd𝑧) = 𝑦))
4123, 40rexlimi 3006 . . . . . 6 (∃𝑥𝐴 𝑦𝐵 → ∃𝑧𝑇 (2nd𝑧) = 𝑦)
4219, 41impbii 198 . . . . 5 (∃𝑧𝑇 (2nd𝑧) = 𝑦 ↔ ∃𝑥𝐴 𝑦𝐵)
43 fvelimab 6163 . . . . . 6 ((2nd Fn V ∧ 𝑇 ⊆ V) → (𝑦 ∈ (2nd𝑇) ↔ ∃𝑧𝑇 (2nd𝑧) = 𝑦))
444, 5, 43mp2an 704 . . . . 5 (𝑦 ∈ (2nd𝑇) ↔ ∃𝑧𝑇 (2nd𝑧) = 𝑦)
45 eliun 4460 . . . . 5 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
4642, 44, 453bitr4i 291 . . . 4 (𝑦 ∈ (2nd𝑇) ↔ 𝑦 𝑥𝐴 𝐵)
4746eqriv 2607 . . 3 (2nd𝑇) = 𝑥𝐴 𝐵
488, 47eqtr3i 2634 . 2 ran (2nd𝑇) = 𝑥𝐴 𝐵
49 df-fo 5810 . 2 ((2nd𝑇):𝑇onto 𝑥𝐴 𝐵 ↔ ((2nd𝑇) Fn 𝑇 ∧ ran (2nd𝑇) = 𝑥𝐴 𝐵))
507, 48, 49mpbir2an 957 1 (2nd𝑇):𝑇onto 𝑥𝐴 𝐵
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  {csn 4125  ⟨cop 4131  ∪ ciun 4455   × cxp 5036  ran crn 5039   ↾ cres 5040   “ cima 5041   Fn wfn 5799  ⟶wf 5800  –onto→wfo 5802  ‘cfv 5804  2nd c2nd 7058 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812  df-2nd 7060 This theorem is referenced by:  iundomg  9242
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