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Theorem founiiun0 43501
Description: Union expressed as an indexed union, when a map onto is given. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Assertion
Ref Expression
founiiun0 (𝐹:𝐴onto→(𝐵 ∪ {∅}) → 𝐵 = 𝑥𝐴 (𝐹𝑥))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹

Proof of Theorem founiiun0
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 uniiun 5022 . 2 𝐵 = 𝑦𝐵 𝑦
2 elun1 4140 . . . . . . 7 (𝑦𝐵𝑦 ∈ (𝐵 ∪ {∅}))
3 foelcdmi 6908 . . . . . . 7 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝑦 ∈ (𝐵 ∪ {∅})) → ∃𝑥𝐴 (𝐹𝑥) = 𝑦)
42, 3sylan2 594 . . . . . 6 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝑦𝐵) → ∃𝑥𝐴 (𝐹𝑥) = 𝑦)
5 eqimss2 4005 . . . . . . 7 ((𝐹𝑥) = 𝑦𝑦 ⊆ (𝐹𝑥))
65reximi 3084 . . . . . 6 (∃𝑥𝐴 (𝐹𝑥) = 𝑦 → ∃𝑥𝐴 𝑦 ⊆ (𝐹𝑥))
74, 6syl 17 . . . . 5 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝑦𝐵) → ∃𝑥𝐴 𝑦 ⊆ (𝐹𝑥))
87ralrimiva 3140 . . . 4 (𝐹:𝐴onto→(𝐵 ∪ {∅}) → ∀𝑦𝐵𝑥𝐴 𝑦 ⊆ (𝐹𝑥))
9 iunss2 5013 . . . 4 (∀𝑦𝐵𝑥𝐴 𝑦 ⊆ (𝐹𝑥) → 𝑦𝐵 𝑦 𝑥𝐴 (𝐹𝑥))
108, 9syl 17 . . 3 (𝐹:𝐴onto→(𝐵 ∪ {∅}) → 𝑦𝐵 𝑦 𝑥𝐴 (𝐹𝑥))
11 simpl 484 . . . . . 6 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝐵 = ∅) → 𝐹:𝐴onto→(𝐵 ∪ {∅}))
12 uneq1 4120 . . . . . . . . 9 (𝐵 = ∅ → (𝐵 ∪ {∅}) = (∅ ∪ {∅}))
13 0un 4356 . . . . . . . . 9 (∅ ∪ {∅}) = {∅}
1412, 13eqtrdi 2789 . . . . . . . 8 (𝐵 = ∅ → (𝐵 ∪ {∅}) = {∅})
1514adantl 483 . . . . . . 7 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝐵 = ∅) → (𝐵 ∪ {∅}) = {∅})
16 foeq3 6758 . . . . . . 7 ((𝐵 ∪ {∅}) = {∅} → (𝐹:𝐴onto→(𝐵 ∪ {∅}) ↔ 𝐹:𝐴onto→{∅}))
1715, 16syl 17 . . . . . 6 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝐵 = ∅) → (𝐹:𝐴onto→(𝐵 ∪ {∅}) ↔ 𝐹:𝐴onto→{∅}))
1811, 17mpbid 231 . . . . 5 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝐵 = ∅) → 𝐹:𝐴onto→{∅})
19 founiiun 43488 . . . . . . 7 (𝐹:𝐴onto→{∅} → {∅} = 𝑥𝐴 (𝐹𝑥))
20 unisn0 43354 . . . . . . 7 {∅} = ∅
2119, 20eqtr3di 2788 . . . . . 6 (𝐹:𝐴onto→{∅} → 𝑥𝐴 (𝐹𝑥) = ∅)
22 0ss 4360 . . . . . 6 ∅ ⊆ 𝑦𝐵 𝑦
2321, 22eqsstrdi 4002 . . . . 5 (𝐹:𝐴onto→{∅} → 𝑥𝐴 (𝐹𝑥) ⊆ 𝑦𝐵 𝑦)
2418, 23syl 17 . . . 4 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝐵 = ∅) → 𝑥𝐴 (𝐹𝑥) ⊆ 𝑦𝐵 𝑦)
25 ssid 3970 . . . . . . . . 9 (𝐹𝑥) ⊆ (𝐹𝑥)
26 sseq2 3974 . . . . . . . . . 10 (𝑦 = (𝐹𝑥) → ((𝐹𝑥) ⊆ 𝑦 ↔ (𝐹𝑥) ⊆ (𝐹𝑥)))
2726rspcev 3583 . . . . . . . . 9 (((𝐹𝑥) ∈ 𝐵 ∧ (𝐹𝑥) ⊆ (𝐹𝑥)) → ∃𝑦𝐵 (𝐹𝑥) ⊆ 𝑦)
2825, 27mpan2 690 . . . . . . . 8 ((𝐹𝑥) ∈ 𝐵 → ∃𝑦𝐵 (𝐹𝑥) ⊆ 𝑦)
2928adantl 483 . . . . . . 7 ((((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) ∧ 𝑥𝐴) ∧ (𝐹𝑥) ∈ 𝐵) → ∃𝑦𝐵 (𝐹𝑥) ⊆ 𝑦)
30 fof 6760 . . . . . . . . . . . 12 (𝐹:𝐴onto→(𝐵 ∪ {∅}) → 𝐹:𝐴⟶(𝐵 ∪ {∅}))
3130ffvelcdmda 7039 . . . . . . . . . . 11 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝑥𝐴) → (𝐹𝑥) ∈ (𝐵 ∪ {∅}))
32 elunnel1 4113 . . . . . . . . . . 11 (((𝐹𝑥) ∈ (𝐵 ∪ {∅}) ∧ ¬ (𝐹𝑥) ∈ 𝐵) → (𝐹𝑥) ∈ {∅})
3331, 32sylan 581 . . . . . . . . . 10 (((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝑥𝐴) ∧ ¬ (𝐹𝑥) ∈ 𝐵) → (𝐹𝑥) ∈ {∅})
34 elsni 4607 . . . . . . . . . 10 ((𝐹𝑥) ∈ {∅} → (𝐹𝑥) = ∅)
3533, 34syl 17 . . . . . . . . 9 (((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝑥𝐴) ∧ ¬ (𝐹𝑥) ∈ 𝐵) → (𝐹𝑥) = ∅)
3635adantllr 718 . . . . . . . 8 ((((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) ∧ 𝑥𝐴) ∧ ¬ (𝐹𝑥) ∈ 𝐵) → (𝐹𝑥) = ∅)
37 neq0 4309 . . . . . . . . . . . . 13 𝐵 = ∅ ↔ ∃𝑦 𝑦𝐵)
3837biimpi 215 . . . . . . . . . . . 12 𝐵 = ∅ → ∃𝑦 𝑦𝐵)
3938adantr 482 . . . . . . . . . . 11 ((¬ 𝐵 = ∅ ∧ (𝐹𝑥) = ∅) → ∃𝑦 𝑦𝐵)
40 id 22 . . . . . . . . . . . . . . . 16 ((𝐹𝑥) = ∅ → (𝐹𝑥) = ∅)
41 0ss 4360 . . . . . . . . . . . . . . . 16 ∅ ⊆ 𝑦
4240, 41eqsstrdi 4002 . . . . . . . . . . . . . . 15 ((𝐹𝑥) = ∅ → (𝐹𝑥) ⊆ 𝑦)
4342anim1ci 617 . . . . . . . . . . . . . 14 (((𝐹𝑥) = ∅ ∧ 𝑦𝐵) → (𝑦𝐵 ∧ (𝐹𝑥) ⊆ 𝑦))
4443ex 414 . . . . . . . . . . . . 13 ((𝐹𝑥) = ∅ → (𝑦𝐵 → (𝑦𝐵 ∧ (𝐹𝑥) ⊆ 𝑦)))
4544adantl 483 . . . . . . . . . . . 12 ((¬ 𝐵 = ∅ ∧ (𝐹𝑥) = ∅) → (𝑦𝐵 → (𝑦𝐵 ∧ (𝐹𝑥) ⊆ 𝑦)))
4645eximdv 1921 . . . . . . . . . . 11 ((¬ 𝐵 = ∅ ∧ (𝐹𝑥) = ∅) → (∃𝑦 𝑦𝐵 → ∃𝑦(𝑦𝐵 ∧ (𝐹𝑥) ⊆ 𝑦)))
4739, 46mpd 15 . . . . . . . . . 10 ((¬ 𝐵 = ∅ ∧ (𝐹𝑥) = ∅) → ∃𝑦(𝑦𝐵 ∧ (𝐹𝑥) ⊆ 𝑦))
48 df-rex 3071 . . . . . . . . . 10 (∃𝑦𝐵 (𝐹𝑥) ⊆ 𝑦 ↔ ∃𝑦(𝑦𝐵 ∧ (𝐹𝑥) ⊆ 𝑦))
4947, 48sylibr 233 . . . . . . . . 9 ((¬ 𝐵 = ∅ ∧ (𝐹𝑥) = ∅) → ∃𝑦𝐵 (𝐹𝑥) ⊆ 𝑦)
5049ad4ant24 753 . . . . . . . 8 ((((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) ∧ 𝑥𝐴) ∧ (𝐹𝑥) = ∅) → ∃𝑦𝐵 (𝐹𝑥) ⊆ 𝑦)
5136, 50syldan 592 . . . . . . 7 ((((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) ∧ 𝑥𝐴) ∧ ¬ (𝐹𝑥) ∈ 𝐵) → ∃𝑦𝐵 (𝐹𝑥) ⊆ 𝑦)
5229, 51pm2.61dan 812 . . . . . 6 (((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) ∧ 𝑥𝐴) → ∃𝑦𝐵 (𝐹𝑥) ⊆ 𝑦)
5352ralrimiva 3140 . . . . 5 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) → ∀𝑥𝐴𝑦𝐵 (𝐹𝑥) ⊆ 𝑦)
54 iunss2 5013 . . . . 5 (∀𝑥𝐴𝑦𝐵 (𝐹𝑥) ⊆ 𝑦 𝑥𝐴 (𝐹𝑥) ⊆ 𝑦𝐵 𝑦)
5553, 54syl 17 . . . 4 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) → 𝑥𝐴 (𝐹𝑥) ⊆ 𝑦𝐵 𝑦)
5624, 55pm2.61dan 812 . . 3 (𝐹:𝐴onto→(𝐵 ∪ {∅}) → 𝑥𝐴 (𝐹𝑥) ⊆ 𝑦𝐵 𝑦)
5710, 56eqssd 3965 . 2 (𝐹:𝐴onto→(𝐵 ∪ {∅}) → 𝑦𝐵 𝑦 = 𝑥𝐴 (𝐹𝑥))
581, 57eqtrid 2785 1 (𝐹:𝐴onto→(𝐵 ∪ {∅}) → 𝐵 = 𝑥𝐴 (𝐹𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397   = wceq 1542  wex 1782  wcel 2107  wral 3061  wrex 3070  cun 3912  wss 3914  c0 4286  {csn 4590   cuni 4869   ciun 4958  ontowfo 6498  cfv 6500
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fo 6506  df-fv 6508
This theorem is referenced by:  ismeannd  44798
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