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Theorem iunfo 9305
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 7134 . . . 4 2nd :V–onto→V
2 fof 6072 . . . 4 (2nd :V–onto→V → 2nd :V⟶V)
3 ffn 6002 . . . 4 (2nd :V⟶V → 2nd Fn V)
41, 2, 3mp2b 10 . . 3 2nd Fn V
5 ssv 3604 . . 3 𝑇 ⊆ V
6 fnssres 5962 . . 3 ((2nd Fn V ∧ 𝑇 ⊆ V) → (2nd𝑇) Fn 𝑇)
74, 5, 6mp2an 707 . 2 (2nd𝑇) Fn 𝑇
8 df-ima 5087 . . 3 (2nd𝑇) = ran (2nd𝑇)
9 iunfo.1 . . . . . . . . . . 11 𝑇 = 𝑥𝐴 ({𝑥} × 𝐵)
109eleq2i 2690 . . . . . . . . . 10 (𝑧𝑇𝑧 𝑥𝐴 ({𝑥} × 𝐵))
11 eliun 4490 . . . . . . . . . 10 (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↔ ∃𝑥𝐴 𝑧 ∈ ({𝑥} × 𝐵))
1210, 11bitri 264 . . . . . . . . 9 (𝑧𝑇 ↔ ∃𝑥𝐴 𝑧 ∈ ({𝑥} × 𝐵))
13 xp2nd 7144 . . . . . . . . . . 11 (𝑧 ∈ ({𝑥} × 𝐵) → (2nd𝑧) ∈ 𝐵)
14 eleq1 2686 . . . . . . . . . . 11 ((2nd𝑧) = 𝑦 → ((2nd𝑧) ∈ 𝐵𝑦𝐵))
1513, 14syl5ib 234 . . . . . . . . . 10 ((2nd𝑧) = 𝑦 → (𝑧 ∈ ({𝑥} × 𝐵) → 𝑦𝐵))
1615reximdv 3010 . . . . . . . . 9 ((2nd𝑧) = 𝑦 → (∃𝑥𝐴 𝑧 ∈ ({𝑥} × 𝐵) → ∃𝑥𝐴 𝑦𝐵))
1712, 16syl5bi 232 . . . . . . . 8 ((2nd𝑧) = 𝑦 → (𝑧𝑇 → ∃𝑥𝐴 𝑦𝐵))
1817impcom 446 . . . . . . 7 ((𝑧𝑇 ∧ (2nd𝑧) = 𝑦) → ∃𝑥𝐴 𝑦𝐵)
1918rexlimiva 3021 . . . . . 6 (∃𝑧𝑇 (2nd𝑧) = 𝑦 → ∃𝑥𝐴 𝑦𝐵)
20 nfiu1 4516 . . . . . . . . 9 𝑥 𝑥𝐴 ({𝑥} × 𝐵)
219, 20nfcxfr 2759 . . . . . . . 8 𝑥𝑇
22 nfv 1840 . . . . . . . 8 𝑥(2nd𝑧) = 𝑦
2321, 22nfrex 3001 . . . . . . 7 𝑥𝑧𝑇 (2nd𝑧) = 𝑦
24 ssiun2 4529 . . . . . . . . . . . 12 (𝑥𝐴 → ({𝑥} × 𝐵) ⊆ 𝑥𝐴 ({𝑥} × 𝐵))
2524adantr 481 . . . . . . . . . . 11 ((𝑥𝐴𝑦𝐵) → ({𝑥} × 𝐵) ⊆ 𝑥𝐴 ({𝑥} × 𝐵))
26 simpr 477 . . . . . . . . . . . 12 ((𝑥𝐴𝑦𝐵) → 𝑦𝐵)
27 vsnid 4180 . . . . . . . . . . . . 13 𝑥 ∈ {𝑥}
28 opelxp 5106 . . . . . . . . . . . . 13 (⟨𝑥, 𝑦⟩ ∈ ({𝑥} × 𝐵) ↔ (𝑥 ∈ {𝑥} ∧ 𝑦𝐵))
2927, 28mpbiran 952 . . . . . . . . . . . 12 (⟨𝑥, 𝑦⟩ ∈ ({𝑥} × 𝐵) ↔ 𝑦𝐵)
3026, 29sylibr 224 . . . . . . . . . . 11 ((𝑥𝐴𝑦𝐵) → ⟨𝑥, 𝑦⟩ ∈ ({𝑥} × 𝐵))
3125, 30sseldd 3584 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐵) → ⟨𝑥, 𝑦⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵))
3231, 9syl6eleqr 2709 . . . . . . . . 9 ((𝑥𝐴𝑦𝐵) → ⟨𝑥, 𝑦⟩ ∈ 𝑇)
33 vex 3189 . . . . . . . . . 10 𝑥 ∈ V
34 vex 3189 . . . . . . . . . 10 𝑦 ∈ V
3533, 34op2nd 7122 . . . . . . . . 9 (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
36 fveq2 6148 . . . . . . . . . . 11 (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd𝑧) = (2nd ‘⟨𝑥, 𝑦⟩))
3736eqeq1d 2623 . . . . . . . . . 10 (𝑧 = ⟨𝑥, 𝑦⟩ → ((2nd𝑧) = 𝑦 ↔ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦))
3837rspcev 3295 . . . . . . . . 9 ((⟨𝑥, 𝑦⟩ ∈ 𝑇 ∧ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦) → ∃𝑧𝑇 (2nd𝑧) = 𝑦)
3932, 35, 38sylancl 693 . . . . . . . 8 ((𝑥𝐴𝑦𝐵) → ∃𝑧𝑇 (2nd𝑧) = 𝑦)
4039ex 450 . . . . . . 7 (𝑥𝐴 → (𝑦𝐵 → ∃𝑧𝑇 (2nd𝑧) = 𝑦))
4123, 40rexlimi 3017 . . . . . 6 (∃𝑥𝐴 𝑦𝐵 → ∃𝑧𝑇 (2nd𝑧) = 𝑦)
4219, 41impbii 199 . . . . 5 (∃𝑧𝑇 (2nd𝑧) = 𝑦 ↔ ∃𝑥𝐴 𝑦𝐵)
43 fvelimab 6210 . . . . . 6 ((2nd Fn V ∧ 𝑇 ⊆ V) → (𝑦 ∈ (2nd𝑇) ↔ ∃𝑧𝑇 (2nd𝑧) = 𝑦))
444, 5, 43mp2an 707 . . . . 5 (𝑦 ∈ (2nd𝑇) ↔ ∃𝑧𝑇 (2nd𝑧) = 𝑦)
45 eliun 4490 . . . . 5 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
4642, 44, 453bitr4i 292 . . . 4 (𝑦 ∈ (2nd𝑇) ↔ 𝑦 𝑥𝐴 𝐵)
4746eqriv 2618 . . 3 (2nd𝑇) = 𝑥𝐴 𝐵
488, 47eqtr3i 2645 . 2 ran (2nd𝑇) = 𝑥𝐴 𝐵
49 df-fo 5853 . 2 ((2nd𝑇):𝑇onto 𝑥𝐴 𝐵 ↔ ((2nd𝑇) Fn 𝑇 ∧ ran (2nd𝑇) = 𝑥𝐴 𝐵))
507, 48, 49mpbir2an 954 1 (2nd𝑇):𝑇onto 𝑥𝐴 𝐵
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wcel 1987  wrex 2908  Vcvv 3186  wss 3555  {csn 4148  cop 4154   ciun 4485   × cxp 5072  ran crn 5075  cres 5076  cima 5077   Fn wfn 5842  wf 5843  ontowfo 5845  cfv 5847  2nd c2nd 7112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fo 5853  df-fv 5855  df-2nd 7114
This theorem is referenced by:  iundomg  9307
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