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Theorem founiiun0 40027
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 4731 . . 3 𝐵 = 𝑦𝐵 𝑦
21a1i 11 . 2 (𝐹:𝐴onto→(𝐵 ∪ {∅}) → 𝐵 = 𝑦𝐵 𝑦)
3 simpl 474 . . . . . . 7 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝑦𝐵) → 𝐹:𝐴onto→(𝐵 ∪ {∅}))
4 elun1 3944 . . . . . . . 8 (𝑦𝐵𝑦 ∈ (𝐵 ∪ {∅}))
54adantl 473 . . . . . . 7 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝑦𝐵) → 𝑦 ∈ (𝐵 ∪ {∅}))
6 foelrni 6435 . . . . . . 7 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝑦 ∈ (𝐵 ∪ {∅})) → ∃𝑥𝐴 (𝐹𝑥) = 𝑦)
73, 5, 6syl2anc 579 . . . . . 6 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝑦𝐵) → ∃𝑥𝐴 (𝐹𝑥) = 𝑦)
8 eqimss2 3820 . . . . . . 7 ((𝐹𝑥) = 𝑦𝑦 ⊆ (𝐹𝑥))
98reximi 3157 . . . . . 6 (∃𝑥𝐴 (𝐹𝑥) = 𝑦 → ∃𝑥𝐴 𝑦 ⊆ (𝐹𝑥))
107, 9syl 17 . . . . 5 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝑦𝐵) → ∃𝑥𝐴 𝑦 ⊆ (𝐹𝑥))
1110ralrimiva 3113 . . . 4 (𝐹:𝐴onto→(𝐵 ∪ {∅}) → ∀𝑦𝐵𝑥𝐴 𝑦 ⊆ (𝐹𝑥))
12 iunss2 4723 . . . 4 (∀𝑦𝐵𝑥𝐴 𝑦 ⊆ (𝐹𝑥) → 𝑦𝐵 𝑦 𝑥𝐴 (𝐹𝑥))
1311, 12syl 17 . . 3 (𝐹:𝐴onto→(𝐵 ∪ {∅}) → 𝑦𝐵 𝑦 𝑥𝐴 (𝐹𝑥))
14 simpl 474 . . . . . 6 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝐵 = ∅) → 𝐹:𝐴onto→(𝐵 ∪ {∅}))
15 uneq1 3924 . . . . . . . . 9 (𝐵 = ∅ → (𝐵 ∪ {∅}) = (∅ ∪ {∅}))
16 0un 39869 . . . . . . . . . 10 (∅ ∪ {∅}) = {∅}
1716a1i 11 . . . . . . . . 9 (𝐵 = ∅ → (∅ ∪ {∅}) = {∅})
1815, 17eqtrd 2799 . . . . . . . 8 (𝐵 = ∅ → (𝐵 ∪ {∅}) = {∅})
1918adantl 473 . . . . . . 7 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝐵 = ∅) → (𝐵 ∪ {∅}) = {∅})
20 foeq3 6298 . . . . . . 7 ((𝐵 ∪ {∅}) = {∅} → (𝐹:𝐴onto→(𝐵 ∪ {∅}) ↔ 𝐹:𝐴onto→{∅}))
2119, 20syl 17 . . . . . 6 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝐵 = ∅) → (𝐹:𝐴onto→(𝐵 ∪ {∅}) ↔ 𝐹:𝐴onto→{∅}))
2214, 21mpbid 223 . . . . 5 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝐵 = ∅) → 𝐹:𝐴onto→{∅})
23 unisn0 39876 . . . . . . . . 9 {∅} = ∅
2423eqcomi 2774 . . . . . . . 8 ∅ = {∅}
2524a1i 11 . . . . . . 7 (𝐹:𝐴onto→{∅} → ∅ = {∅})
26 founiiun 40010 . . . . . . 7 (𝐹:𝐴onto→{∅} → {∅} = 𝑥𝐴 (𝐹𝑥))
2725, 26eqtr2d 2800 . . . . . 6 (𝐹:𝐴onto→{∅} → 𝑥𝐴 (𝐹𝑥) = ∅)
28 0ss 4136 . . . . . . 7 ∅ ⊆ 𝑦𝐵 𝑦
2928a1i 11 . . . . . 6 (𝐹:𝐴onto→{∅} → ∅ ⊆ 𝑦𝐵 𝑦)
3027, 29eqsstrd 3801 . . . . 5 (𝐹:𝐴onto→{∅} → 𝑥𝐴 (𝐹𝑥) ⊆ 𝑦𝐵 𝑦)
3122, 30syl 17 . . . 4 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝐵 = ∅) → 𝑥𝐴 (𝐹𝑥) ⊆ 𝑦𝐵 𝑦)
32 ssid 3785 . . . . . . . . 9 (𝐹𝑥) ⊆ (𝐹𝑥)
33 sseq2 3789 . . . . . . . . . 10 (𝑦 = (𝐹𝑥) → ((𝐹𝑥) ⊆ 𝑦 ↔ (𝐹𝑥) ⊆ (𝐹𝑥)))
3433rspcev 3462 . . . . . . . . 9 (((𝐹𝑥) ∈ 𝐵 ∧ (𝐹𝑥) ⊆ (𝐹𝑥)) → ∃𝑦𝐵 (𝐹𝑥) ⊆ 𝑦)
3532, 34mpan2 682 . . . . . . . 8 ((𝐹𝑥) ∈ 𝐵 → ∃𝑦𝐵 (𝐹𝑥) ⊆ 𝑦)
3635adantl 473 . . . . . . 7 ((((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) ∧ 𝑥𝐴) ∧ (𝐹𝑥) ∈ 𝐵) → ∃𝑦𝐵 (𝐹𝑥) ⊆ 𝑦)
37 simpl 474 . . . . . . . 8 ((((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) ∧ 𝑥𝐴) ∧ ¬ (𝐹𝑥) ∈ 𝐵) → ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) ∧ 𝑥𝐴))
38 fof 6300 . . . . . . . . . . . . 13 (𝐹:𝐴onto→(𝐵 ∪ {∅}) → 𝐹:𝐴⟶(𝐵 ∪ {∅}))
3938ffvelrnda 6551 . . . . . . . . . . . 12 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝑥𝐴) → (𝐹𝑥) ∈ (𝐵 ∪ {∅}))
4039adantr 472 . . . . . . . . . . 11 (((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝑥𝐴) ∧ ¬ (𝐹𝑥) ∈ 𝐵) → (𝐹𝑥) ∈ (𝐵 ∪ {∅}))
41 simpr 477 . . . . . . . . . . 11 (((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝑥𝐴) ∧ ¬ (𝐹𝑥) ∈ 𝐵) → ¬ (𝐹𝑥) ∈ 𝐵)
42 elunnel1 3918 . . . . . . . . . . 11 (((𝐹𝑥) ∈ (𝐵 ∪ {∅}) ∧ ¬ (𝐹𝑥) ∈ 𝐵) → (𝐹𝑥) ∈ {∅})
4340, 41, 42syl2anc 579 . . . . . . . . . 10 (((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝑥𝐴) ∧ ¬ (𝐹𝑥) ∈ 𝐵) → (𝐹𝑥) ∈ {∅})
44 elsni 4353 . . . . . . . . . 10 ((𝐹𝑥) ∈ {∅} → (𝐹𝑥) = ∅)
4543, 44syl 17 . . . . . . . . 9 (((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝑥𝐴) ∧ ¬ (𝐹𝑥) ∈ 𝐵) → (𝐹𝑥) = ∅)
4645adantllr 710 . . . . . . . 8 ((((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) ∧ 𝑥𝐴) ∧ ¬ (𝐹𝑥) ∈ 𝐵) → (𝐹𝑥) = ∅)
47 neq0 4096 . . . . . . . . . . . . 13 𝐵 = ∅ ↔ ∃𝑦 𝑦𝐵)
4847biimpi 207 . . . . . . . . . . . 12 𝐵 = ∅ → ∃𝑦 𝑦𝐵)
4948adantr 472 . . . . . . . . . . 11 ((¬ 𝐵 = ∅ ∧ (𝐹𝑥) = ∅) → ∃𝑦 𝑦𝐵)
50 simpr 477 . . . . . . . . . . . . . . 15 (((𝐹𝑥) = ∅ ∧ 𝑦𝐵) → 𝑦𝐵)
51 id 22 . . . . . . . . . . . . . . . . 17 ((𝐹𝑥) = ∅ → (𝐹𝑥) = ∅)
52 0ss 4136 . . . . . . . . . . . . . . . . . 18 ∅ ⊆ 𝑦
5352a1i 11 . . . . . . . . . . . . . . . . 17 ((𝐹𝑥) = ∅ → ∅ ⊆ 𝑦)
5451, 53eqsstrd 3801 . . . . . . . . . . . . . . . 16 ((𝐹𝑥) = ∅ → (𝐹𝑥) ⊆ 𝑦)
5554adantr 472 . . . . . . . . . . . . . . 15 (((𝐹𝑥) = ∅ ∧ 𝑦𝐵) → (𝐹𝑥) ⊆ 𝑦)
5650, 55jca 507 . . . . . . . . . . . . . 14 (((𝐹𝑥) = ∅ ∧ 𝑦𝐵) → (𝑦𝐵 ∧ (𝐹𝑥) ⊆ 𝑦))
5756ex 401 . . . . . . . . . . . . 13 ((𝐹𝑥) = ∅ → (𝑦𝐵 → (𝑦𝐵 ∧ (𝐹𝑥) ⊆ 𝑦)))
5857adantl 473 . . . . . . . . . . . 12 ((¬ 𝐵 = ∅ ∧ (𝐹𝑥) = ∅) → (𝑦𝐵 → (𝑦𝐵 ∧ (𝐹𝑥) ⊆ 𝑦)))
5958eximdv 2012 . . . . . . . . . . 11 ((¬ 𝐵 = ∅ ∧ (𝐹𝑥) = ∅) → (∃𝑦 𝑦𝐵 → ∃𝑦(𝑦𝐵 ∧ (𝐹𝑥) ⊆ 𝑦)))
6049, 59mpd 15 . . . . . . . . . 10 ((¬ 𝐵 = ∅ ∧ (𝐹𝑥) = ∅) → ∃𝑦(𝑦𝐵 ∧ (𝐹𝑥) ⊆ 𝑦))
61 df-rex 3061 . . . . . . . . . 10 (∃𝑦𝐵 (𝐹𝑥) ⊆ 𝑦 ↔ ∃𝑦(𝑦𝐵 ∧ (𝐹𝑥) ⊆ 𝑦))
6260, 61sylibr 225 . . . . . . . . 9 ((¬ 𝐵 = ∅ ∧ (𝐹𝑥) = ∅) → ∃𝑦𝐵 (𝐹𝑥) ⊆ 𝑦)
6362ad4ant24 763 . . . . . . . 8 ((((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) ∧ 𝑥𝐴) ∧ (𝐹𝑥) = ∅) → ∃𝑦𝐵 (𝐹𝑥) ⊆ 𝑦)
6437, 46, 63syl2anc 579 . . . . . . 7 ((((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) ∧ 𝑥𝐴) ∧ ¬ (𝐹𝑥) ∈ 𝐵) → ∃𝑦𝐵 (𝐹𝑥) ⊆ 𝑦)
6536, 64pm2.61dan 847 . . . . . 6 (((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) ∧ 𝑥𝐴) → ∃𝑦𝐵 (𝐹𝑥) ⊆ 𝑦)
6665ralrimiva 3113 . . . . 5 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) → ∀𝑥𝐴𝑦𝐵 (𝐹𝑥) ⊆ 𝑦)
67 iunss2 4723 . . . . 5 (∀𝑥𝐴𝑦𝐵 (𝐹𝑥) ⊆ 𝑦 𝑥𝐴 (𝐹𝑥) ⊆ 𝑦𝐵 𝑦)
6866, 67syl 17 . . . 4 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) → 𝑥𝐴 (𝐹𝑥) ⊆ 𝑦𝐵 𝑦)
6931, 68pm2.61dan 847 . . 3 (𝐹:𝐴onto→(𝐵 ∪ {∅}) → 𝑥𝐴 (𝐹𝑥) ⊆ 𝑦𝐵 𝑦)
7013, 69eqssd 3780 . 2 (𝐹:𝐴onto→(𝐵 ∪ {∅}) → 𝑦𝐵 𝑦 = 𝑥𝐴 (𝐹𝑥))
712, 70eqtrd 2799 1 (𝐹:𝐴onto→(𝐵 ∪ {∅}) → 𝐵 = 𝑥𝐴 (𝐹𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384   = wceq 1652  wex 1874  wcel 2155  wral 3055  wrex 3056  cun 3732  wss 3734  c0 4081  {csn 4336   cuni 4596   ciun 4678  ontowfo 6068  cfv 6070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pr 5064
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-fo 6076  df-fv 6078
This theorem is referenced by:  ismeannd  41324
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