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Theorem founiiun0 39691
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 4605 . . 3 𝐵 = 𝑦𝐵 𝑦
21a1i 11 . 2 (𝐹:𝐴onto→(𝐵 ∪ {∅}) → 𝐵 = 𝑦𝐵 𝑦)
3 simpl 472 . . . . . . 7 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝑦𝐵) → 𝐹:𝐴onto→(𝐵 ∪ {∅}))
4 elun1 3813 . . . . . . . 8 (𝑦𝐵𝑦 ∈ (𝐵 ∪ {∅}))
54adantl 481 . . . . . . 7 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝑦𝐵) → 𝑦 ∈ (𝐵 ∪ {∅}))
6 foelrni 6283 . . . . . . 7 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝑦 ∈ (𝐵 ∪ {∅})) → ∃𝑥𝐴 (𝐹𝑥) = 𝑦)
73, 5, 6syl2anc 694 . . . . . 6 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝑦𝐵) → ∃𝑥𝐴 (𝐹𝑥) = 𝑦)
8 eqimss2 3691 . . . . . . 7 ((𝐹𝑥) = 𝑦𝑦 ⊆ (𝐹𝑥))
98reximi 3040 . . . . . 6 (∃𝑥𝐴 (𝐹𝑥) = 𝑦 → ∃𝑥𝐴 𝑦 ⊆ (𝐹𝑥))
107, 9syl 17 . . . . 5 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝑦𝐵) → ∃𝑥𝐴 𝑦 ⊆ (𝐹𝑥))
1110ralrimiva 2995 . . . 4 (𝐹:𝐴onto→(𝐵 ∪ {∅}) → ∀𝑦𝐵𝑥𝐴 𝑦 ⊆ (𝐹𝑥))
12 iunss2 4597 . . . 4 (∀𝑦𝐵𝑥𝐴 𝑦 ⊆ (𝐹𝑥) → 𝑦𝐵 𝑦 𝑥𝐴 (𝐹𝑥))
1311, 12syl 17 . . 3 (𝐹:𝐴onto→(𝐵 ∪ {∅}) → 𝑦𝐵 𝑦 𝑥𝐴 (𝐹𝑥))
14 simpl 472 . . . . . 6 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝐵 = ∅) → 𝐹:𝐴onto→(𝐵 ∪ {∅}))
15 uneq1 3793 . . . . . . . . 9 (𝐵 = ∅ → (𝐵 ∪ {∅}) = (∅ ∪ {∅}))
16 0un 39529 . . . . . . . . . 10 (∅ ∪ {∅}) = {∅}
1716a1i 11 . . . . . . . . 9 (𝐵 = ∅ → (∅ ∪ {∅}) = {∅})
1815, 17eqtrd 2685 . . . . . . . 8 (𝐵 = ∅ → (𝐵 ∪ {∅}) = {∅})
1918adantl 481 . . . . . . 7 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝐵 = ∅) → (𝐵 ∪ {∅}) = {∅})
20 foeq3 6151 . . . . . . 7 ((𝐵 ∪ {∅}) = {∅} → (𝐹:𝐴onto→(𝐵 ∪ {∅}) ↔ 𝐹:𝐴onto→{∅}))
2119, 20syl 17 . . . . . 6 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝐵 = ∅) → (𝐹:𝐴onto→(𝐵 ∪ {∅}) ↔ 𝐹:𝐴onto→{∅}))
2214, 21mpbid 222 . . . . 5 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝐵 = ∅) → 𝐹:𝐴onto→{∅})
23 unisn0 39536 . . . . . . . . 9 {∅} = ∅
2423eqcomi 2660 . . . . . . . 8 ∅ = {∅}
2524a1i 11 . . . . . . 7 (𝐹:𝐴onto→{∅} → ∅ = {∅})
26 founiiun 39674 . . . . . . 7 (𝐹:𝐴onto→{∅} → {∅} = 𝑥𝐴 (𝐹𝑥))
2725, 26eqtr2d 2686 . . . . . 6 (𝐹:𝐴onto→{∅} → 𝑥𝐴 (𝐹𝑥) = ∅)
28 0ss 4005 . . . . . . 7 ∅ ⊆ 𝑦𝐵 𝑦
2928a1i 11 . . . . . 6 (𝐹:𝐴onto→{∅} → ∅ ⊆ 𝑦𝐵 𝑦)
3027, 29eqsstrd 3672 . . . . 5 (𝐹:𝐴onto→{∅} → 𝑥𝐴 (𝐹𝑥) ⊆ 𝑦𝐵 𝑦)
3122, 30syl 17 . . . 4 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝐵 = ∅) → 𝑥𝐴 (𝐹𝑥) ⊆ 𝑦𝐵 𝑦)
32 ssid 3657 . . . . . . . . 9 (𝐹𝑥) ⊆ (𝐹𝑥)
33 sseq2 3660 . . . . . . . . . 10 (𝑦 = (𝐹𝑥) → ((𝐹𝑥) ⊆ 𝑦 ↔ (𝐹𝑥) ⊆ (𝐹𝑥)))
3433rspcev 3340 . . . . . . . . 9 (((𝐹𝑥) ∈ 𝐵 ∧ (𝐹𝑥) ⊆ (𝐹𝑥)) → ∃𝑦𝐵 (𝐹𝑥) ⊆ 𝑦)
3532, 34mpan2 707 . . . . . . . 8 ((𝐹𝑥) ∈ 𝐵 → ∃𝑦𝐵 (𝐹𝑥) ⊆ 𝑦)
3635adantl 481 . . . . . . 7 ((((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) ∧ 𝑥𝐴) ∧ (𝐹𝑥) ∈ 𝐵) → ∃𝑦𝐵 (𝐹𝑥) ⊆ 𝑦)
37 simpl 472 . . . . . . . 8 ((((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) ∧ 𝑥𝐴) ∧ ¬ (𝐹𝑥) ∈ 𝐵) → ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) ∧ 𝑥𝐴))
38 fof 6153 . . . . . . . . . . . . 13 (𝐹:𝐴onto→(𝐵 ∪ {∅}) → 𝐹:𝐴⟶(𝐵 ∪ {∅}))
3938ffvelrnda 6399 . . . . . . . . . . . 12 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝑥𝐴) → (𝐹𝑥) ∈ (𝐵 ∪ {∅}))
4039adantr 480 . . . . . . . . . . 11 (((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝑥𝐴) ∧ ¬ (𝐹𝑥) ∈ 𝐵) → (𝐹𝑥) ∈ (𝐵 ∪ {∅}))
41 simpr 476 . . . . . . . . . . 11 (((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝑥𝐴) ∧ ¬ (𝐹𝑥) ∈ 𝐵) → ¬ (𝐹𝑥) ∈ 𝐵)
42 elunnel1 3787 . . . . . . . . . . 11 (((𝐹𝑥) ∈ (𝐵 ∪ {∅}) ∧ ¬ (𝐹𝑥) ∈ 𝐵) → (𝐹𝑥) ∈ {∅})
4340, 41, 42syl2anc 694 . . . . . . . . . 10 (((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝑥𝐴) ∧ ¬ (𝐹𝑥) ∈ 𝐵) → (𝐹𝑥) ∈ {∅})
44 elsni 4227 . . . . . . . . . 10 ((𝐹𝑥) ∈ {∅} → (𝐹𝑥) = ∅)
4543, 44syl 17 . . . . . . . . 9 (((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝑥𝐴) ∧ ¬ (𝐹𝑥) ∈ 𝐵) → (𝐹𝑥) = ∅)
4645adantllr 755 . . . . . . . 8 ((((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) ∧ 𝑥𝐴) ∧ ¬ (𝐹𝑥) ∈ 𝐵) → (𝐹𝑥) = ∅)
47 neq0 3963 . . . . . . . . . . . . 13 𝐵 = ∅ ↔ ∃𝑦 𝑦𝐵)
4847biimpi 206 . . . . . . . . . . . 12 𝐵 = ∅ → ∃𝑦 𝑦𝐵)
4948adantr 480 . . . . . . . . . . 11 ((¬ 𝐵 = ∅ ∧ (𝐹𝑥) = ∅) → ∃𝑦 𝑦𝐵)
50 simpr 476 . . . . . . . . . . . . . . 15 (((𝐹𝑥) = ∅ ∧ 𝑦𝐵) → 𝑦𝐵)
51 id 22 . . . . . . . . . . . . . . . . 17 ((𝐹𝑥) = ∅ → (𝐹𝑥) = ∅)
52 0ss 4005 . . . . . . . . . . . . . . . . . 18 ∅ ⊆ 𝑦
5352a1i 11 . . . . . . . . . . . . . . . . 17 ((𝐹𝑥) = ∅ → ∅ ⊆ 𝑦)
5451, 53eqsstrd 3672 . . . . . . . . . . . . . . . 16 ((𝐹𝑥) = ∅ → (𝐹𝑥) ⊆ 𝑦)
5554adantr 480 . . . . . . . . . . . . . . 15 (((𝐹𝑥) = ∅ ∧ 𝑦𝐵) → (𝐹𝑥) ⊆ 𝑦)
5650, 55jca 553 . . . . . . . . . . . . . 14 (((𝐹𝑥) = ∅ ∧ 𝑦𝐵) → (𝑦𝐵 ∧ (𝐹𝑥) ⊆ 𝑦))
5756ex 449 . . . . . . . . . . . . 13 ((𝐹𝑥) = ∅ → (𝑦𝐵 → (𝑦𝐵 ∧ (𝐹𝑥) ⊆ 𝑦)))
5857adantl 481 . . . . . . . . . . . 12 ((¬ 𝐵 = ∅ ∧ (𝐹𝑥) = ∅) → (𝑦𝐵 → (𝑦𝐵 ∧ (𝐹𝑥) ⊆ 𝑦)))
5958eximdv 1886 . . . . . . . . . . 11 ((¬ 𝐵 = ∅ ∧ (𝐹𝑥) = ∅) → (∃𝑦 𝑦𝐵 → ∃𝑦(𝑦𝐵 ∧ (𝐹𝑥) ⊆ 𝑦)))
6049, 59mpd 15 . . . . . . . . . 10 ((¬ 𝐵 = ∅ ∧ (𝐹𝑥) = ∅) → ∃𝑦(𝑦𝐵 ∧ (𝐹𝑥) ⊆ 𝑦))
61 df-rex 2947 . . . . . . . . . 10 (∃𝑦𝐵 (𝐹𝑥) ⊆ 𝑦 ↔ ∃𝑦(𝑦𝐵 ∧ (𝐹𝑥) ⊆ 𝑦))
6260, 61sylibr 224 . . . . . . . . 9 ((¬ 𝐵 = ∅ ∧ (𝐹𝑥) = ∅) → ∃𝑦𝐵 (𝐹𝑥) ⊆ 𝑦)
6362ad4ant24 1327 . . . . . . . 8 ((((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) ∧ 𝑥𝐴) ∧ (𝐹𝑥) = ∅) → ∃𝑦𝐵 (𝐹𝑥) ⊆ 𝑦)
6437, 46, 63syl2anc 694 . . . . . . 7 ((((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) ∧ 𝑥𝐴) ∧ ¬ (𝐹𝑥) ∈ 𝐵) → ∃𝑦𝐵 (𝐹𝑥) ⊆ 𝑦)
6536, 64pm2.61dan 849 . . . . . 6 (((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) ∧ 𝑥𝐴) → ∃𝑦𝐵 (𝐹𝑥) ⊆ 𝑦)
6665ralrimiva 2995 . . . . 5 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) → ∀𝑥𝐴𝑦𝐵 (𝐹𝑥) ⊆ 𝑦)
67 iunss2 4597 . . . . 5 (∀𝑥𝐴𝑦𝐵 (𝐹𝑥) ⊆ 𝑦 𝑥𝐴 (𝐹𝑥) ⊆ 𝑦𝐵 𝑦)
6866, 67syl 17 . . . 4 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) → 𝑥𝐴 (𝐹𝑥) ⊆ 𝑦𝐵 𝑦)
6931, 68pm2.61dan 849 . . 3 (𝐹:𝐴onto→(𝐵 ∪ {∅}) → 𝑥𝐴 (𝐹𝑥) ⊆ 𝑦𝐵 𝑦)
7013, 69eqssd 3653 . 2 (𝐹:𝐴onto→(𝐵 ∪ {∅}) → 𝑦𝐵 𝑦 = 𝑥𝐴 (𝐹𝑥))
712, 70eqtrd 2685 1 (𝐹:𝐴onto→(𝐵 ∪ {∅}) → 𝐵 = 𝑥𝐴 (𝐹𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1523  wex 1744  wcel 2030  wral 2941  wrex 2942  cun 3605  wss 3607  c0 3948  {csn 4210   cuni 4468   ciun 4552  ontowfo 5924  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fo 5932  df-fv 5934
This theorem is referenced by:  ismeannd  41002
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