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Theorem iunfo 9551
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 7352 . . . 4 2nd :V–onto→V
2 fof 6274 . . . 4 (2nd :V–onto→V → 2nd :V⟶V)
3 ffn 6204 . . . 4 (2nd :V⟶V → 2nd Fn V)
41, 2, 3mp2b 10 . . 3 2nd Fn V
5 ssv 3764 . . 3 𝑇 ⊆ V
6 fnssres 6163 . . 3 ((2nd Fn V ∧ 𝑇 ⊆ V) → (2nd𝑇) Fn 𝑇)
74, 5, 6mp2an 710 . 2 (2nd𝑇) Fn 𝑇
8 df-ima 5277 . . 3 (2nd𝑇) = ran (2nd𝑇)
9 iunfo.1 . . . . . . . . . . 11 𝑇 = 𝑥𝐴 ({𝑥} × 𝐵)
109eleq2i 2829 . . . . . . . . . 10 (𝑧𝑇𝑧 𝑥𝐴 ({𝑥} × 𝐵))
11 eliun 4674 . . . . . . . . . 10 (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↔ ∃𝑥𝐴 𝑧 ∈ ({𝑥} × 𝐵))
1210, 11bitri 264 . . . . . . . . 9 (𝑧𝑇 ↔ ∃𝑥𝐴 𝑧 ∈ ({𝑥} × 𝐵))
13 xp2nd 7364 . . . . . . . . . . 11 (𝑧 ∈ ({𝑥} × 𝐵) → (2nd𝑧) ∈ 𝐵)
14 eleq1 2825 . . . . . . . . . . 11 ((2nd𝑧) = 𝑦 → ((2nd𝑧) ∈ 𝐵𝑦𝐵))
1513, 14syl5ib 234 . . . . . . . . . 10 ((2nd𝑧) = 𝑦 → (𝑧 ∈ ({𝑥} × 𝐵) → 𝑦𝐵))
1615reximdv 3152 . . . . . . . . 9 ((2nd𝑧) = 𝑦 → (∃𝑥𝐴 𝑧 ∈ ({𝑥} × 𝐵) → ∃𝑥𝐴 𝑦𝐵))
1712, 16syl5bi 232 . . . . . . . 8 ((2nd𝑧) = 𝑦 → (𝑧𝑇 → ∃𝑥𝐴 𝑦𝐵))
1817impcom 445 . . . . . . 7 ((𝑧𝑇 ∧ (2nd𝑧) = 𝑦) → ∃𝑥𝐴 𝑦𝐵)
1918rexlimiva 3164 . . . . . 6 (∃𝑧𝑇 (2nd𝑧) = 𝑦 → ∃𝑥𝐴 𝑦𝐵)
20 nfiu1 4700 . . . . . . . . 9 𝑥 𝑥𝐴 ({𝑥} × 𝐵)
219, 20nfcxfr 2898 . . . . . . . 8 𝑥𝑇
22 nfv 1990 . . . . . . . 8 𝑥(2nd𝑧) = 𝑦
2321, 22nfrex 3143 . . . . . . 7 𝑥𝑧𝑇 (2nd𝑧) = 𝑦
24 ssiun2 4713 . . . . . . . . . . . 12 (𝑥𝐴 → ({𝑥} × 𝐵) ⊆ 𝑥𝐴 ({𝑥} × 𝐵))
2524adantr 472 . . . . . . . . . . 11 ((𝑥𝐴𝑦𝐵) → ({𝑥} × 𝐵) ⊆ 𝑥𝐴 ({𝑥} × 𝐵))
26 simpr 479 . . . . . . . . . . . 12 ((𝑥𝐴𝑦𝐵) → 𝑦𝐵)
27 vsnid 4352 . . . . . . . . . . . . 13 𝑥 ∈ {𝑥}
28 opelxp 5301 . . . . . . . . . . . . 13 (⟨𝑥, 𝑦⟩ ∈ ({𝑥} × 𝐵) ↔ (𝑥 ∈ {𝑥} ∧ 𝑦𝐵))
2927, 28mpbiran 991 . . . . . . . . . . . 12 (⟨𝑥, 𝑦⟩ ∈ ({𝑥} × 𝐵) ↔ 𝑦𝐵)
3026, 29sylibr 224 . . . . . . . . . . 11 ((𝑥𝐴𝑦𝐵) → ⟨𝑥, 𝑦⟩ ∈ ({𝑥} × 𝐵))
3125, 30sseldd 3743 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐵) → ⟨𝑥, 𝑦⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵))
3231, 9syl6eleqr 2848 . . . . . . . . 9 ((𝑥𝐴𝑦𝐵) → ⟨𝑥, 𝑦⟩ ∈ 𝑇)
33 vex 3341 . . . . . . . . . 10 𝑥 ∈ V
34 vex 3341 . . . . . . . . . 10 𝑦 ∈ V
3533, 34op2nd 7340 . . . . . . . . 9 (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
36 fveq2 6350 . . . . . . . . . . 11 (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd𝑧) = (2nd ‘⟨𝑥, 𝑦⟩))
3736eqeq1d 2760 . . . . . . . . . 10 (𝑧 = ⟨𝑥, 𝑦⟩ → ((2nd𝑧) = 𝑦 ↔ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦))
3837rspcev 3447 . . . . . . . . 9 ((⟨𝑥, 𝑦⟩ ∈ 𝑇 ∧ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦) → ∃𝑧𝑇 (2nd𝑧) = 𝑦)
3932, 35, 38sylancl 697 . . . . . . . 8 ((𝑥𝐴𝑦𝐵) → ∃𝑧𝑇 (2nd𝑧) = 𝑦)
4039ex 449 . . . . . . 7 (𝑥𝐴 → (𝑦𝐵 → ∃𝑧𝑇 (2nd𝑧) = 𝑦))
4123, 40rexlimi 3160 . . . . . 6 (∃𝑥𝐴 𝑦𝐵 → ∃𝑧𝑇 (2nd𝑧) = 𝑦)
4219, 41impbii 199 . . . . 5 (∃𝑧𝑇 (2nd𝑧) = 𝑦 ↔ ∃𝑥𝐴 𝑦𝐵)
43 fvelimab 6413 . . . . . 6 ((2nd Fn V ∧ 𝑇 ⊆ V) → (𝑦 ∈ (2nd𝑇) ↔ ∃𝑧𝑇 (2nd𝑧) = 𝑦))
444, 5, 43mp2an 710 . . . . 5 (𝑦 ∈ (2nd𝑇) ↔ ∃𝑧𝑇 (2nd𝑧) = 𝑦)
45 eliun 4674 . . . . 5 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
4642, 44, 453bitr4i 292 . . . 4 (𝑦 ∈ (2nd𝑇) ↔ 𝑦 𝑥𝐴 𝐵)
4746eqriv 2755 . . 3 (2nd𝑇) = 𝑥𝐴 𝐵
488, 47eqtr3i 2782 . 2 ran (2nd𝑇) = 𝑥𝐴 𝐵
49 df-fo 6053 . 2 ((2nd𝑇):𝑇onto 𝑥𝐴 𝐵 ↔ ((2nd𝑇) Fn 𝑇 ∧ ran (2nd𝑇) = 𝑥𝐴 𝐵))
507, 48, 49mpbir2an 993 1 (2nd𝑇):𝑇onto 𝑥𝐴 𝐵
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1630  wcel 2137  wrex 3049  Vcvv 3338  wss 3713  {csn 4319  cop 4325   ciun 4670   × cxp 5262  ran crn 5265  cres 5266  cima 5267   Fn wfn 6042  wf 6043  ontowfo 6045  cfv 6047  2nd c2nd 7330
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ral 3053  df-rex 3054  df-rab 3057  df-v 3340  df-sbc 3575  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-id 5172  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-fo 6053  df-fv 6055  df-2nd 7332
This theorem is referenced by:  iundomg  9553
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