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Theorem founiiun0 42307
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 4954 . 2 𝐵 = 𝑦𝐵 𝑦
2 elun1 4076 . . . . . . 7 (𝑦𝐵𝑦 ∈ (𝐵 ∪ {∅}))
3 foelrni 6744 . . . . . . 7 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝑦 ∈ (𝐵 ∪ {∅})) → ∃𝑥𝐴 (𝐹𝑥) = 𝑦)
42, 3sylan2 596 . . . . . 6 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝑦𝐵) → ∃𝑥𝐴 (𝐹𝑥) = 𝑦)
5 eqimss2 3944 . . . . . . 7 ((𝐹𝑥) = 𝑦𝑦 ⊆ (𝐹𝑥))
65reximi 3158 . . . . . 6 (∃𝑥𝐴 (𝐹𝑥) = 𝑦 → ∃𝑥𝐴 𝑦 ⊆ (𝐹𝑥))
74, 6syl 17 . . . . 5 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝑦𝐵) → ∃𝑥𝐴 𝑦 ⊆ (𝐹𝑥))
87ralrimiva 3097 . . . 4 (𝐹:𝐴onto→(𝐵 ∪ {∅}) → ∀𝑦𝐵𝑥𝐴 𝑦 ⊆ (𝐹𝑥))
9 iunss2 4945 . . . 4 (∀𝑦𝐵𝑥𝐴 𝑦 ⊆ (𝐹𝑥) → 𝑦𝐵 𝑦 𝑥𝐴 (𝐹𝑥))
108, 9syl 17 . . 3 (𝐹:𝐴onto→(𝐵 ∪ {∅}) → 𝑦𝐵 𝑦 𝑥𝐴 (𝐹𝑥))
11 simpl 486 . . . . . 6 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝐵 = ∅) → 𝐹:𝐴onto→(𝐵 ∪ {∅}))
12 uneq1 4056 . . . . . . . . 9 (𝐵 = ∅ → (𝐵 ∪ {∅}) = (∅ ∪ {∅}))
13 0un 4291 . . . . . . . . 9 (∅ ∪ {∅}) = {∅}
1412, 13eqtrdi 2790 . . . . . . . 8 (𝐵 = ∅ → (𝐵 ∪ {∅}) = {∅})
1514adantl 485 . . . . . . 7 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝐵 = ∅) → (𝐵 ∪ {∅}) = {∅})
16 foeq3 6601 . . . . . . 7 ((𝐵 ∪ {∅}) = {∅} → (𝐹:𝐴onto→(𝐵 ∪ {∅}) ↔ 𝐹:𝐴onto→{∅}))
1715, 16syl 17 . . . . . 6 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝐵 = ∅) → (𝐹:𝐴onto→(𝐵 ∪ {∅}) ↔ 𝐹:𝐴onto→{∅}))
1811, 17mpbid 235 . . . . 5 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝐵 = ∅) → 𝐹:𝐴onto→{∅})
19 founiiun 42294 . . . . . . 7 (𝐹:𝐴onto→{∅} → {∅} = 𝑥𝐴 (𝐹𝑥))
20 unisn0 42181 . . . . . . 7 {∅} = ∅
2119, 20eqtr3di 2789 . . . . . 6 (𝐹:𝐴onto→{∅} → 𝑥𝐴 (𝐹𝑥) = ∅)
22 0ss 4295 . . . . . 6 ∅ ⊆ 𝑦𝐵 𝑦
2321, 22eqsstrdi 3941 . . . . 5 (𝐹:𝐴onto→{∅} → 𝑥𝐴 (𝐹𝑥) ⊆ 𝑦𝐵 𝑦)
2418, 23syl 17 . . . 4 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝐵 = ∅) → 𝑥𝐴 (𝐹𝑥) ⊆ 𝑦𝐵 𝑦)
25 ssid 3909 . . . . . . . . 9 (𝐹𝑥) ⊆ (𝐹𝑥)
26 sseq2 3913 . . . . . . . . . 10 (𝑦 = (𝐹𝑥) → ((𝐹𝑥) ⊆ 𝑦 ↔ (𝐹𝑥) ⊆ (𝐹𝑥)))
2726rspcev 3529 . . . . . . . . 9 (((𝐹𝑥) ∈ 𝐵 ∧ (𝐹𝑥) ⊆ (𝐹𝑥)) → ∃𝑦𝐵 (𝐹𝑥) ⊆ 𝑦)
2825, 27mpan2 691 . . . . . . . 8 ((𝐹𝑥) ∈ 𝐵 → ∃𝑦𝐵 (𝐹𝑥) ⊆ 𝑦)
2928adantl 485 . . . . . . 7 ((((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) ∧ 𝑥𝐴) ∧ (𝐹𝑥) ∈ 𝐵) → ∃𝑦𝐵 (𝐹𝑥) ⊆ 𝑦)
30 fof 6603 . . . . . . . . . . . 12 (𝐹:𝐴onto→(𝐵 ∪ {∅}) → 𝐹:𝐴⟶(𝐵 ∪ {∅}))
3130ffvelrnda 6874 . . . . . . . . . . 11 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝑥𝐴) → (𝐹𝑥) ∈ (𝐵 ∪ {∅}))
32 elunnel1 4050 . . . . . . . . . . 11 (((𝐹𝑥) ∈ (𝐵 ∪ {∅}) ∧ ¬ (𝐹𝑥) ∈ 𝐵) → (𝐹𝑥) ∈ {∅})
3331, 32sylan 583 . . . . . . . . . 10 (((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝑥𝐴) ∧ ¬ (𝐹𝑥) ∈ 𝐵) → (𝐹𝑥) ∈ {∅})
34 elsni 4543 . . . . . . . . . 10 ((𝐹𝑥) ∈ {∅} → (𝐹𝑥) = ∅)
3533, 34syl 17 . . . . . . . . 9 (((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ 𝑥𝐴) ∧ ¬ (𝐹𝑥) ∈ 𝐵) → (𝐹𝑥) = ∅)
3635adantllr 719 . . . . . . . 8 ((((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) ∧ 𝑥𝐴) ∧ ¬ (𝐹𝑥) ∈ 𝐵) → (𝐹𝑥) = ∅)
37 neq0 4244 . . . . . . . . . . . . 13 𝐵 = ∅ ↔ ∃𝑦 𝑦𝐵)
3837biimpi 219 . . . . . . . . . . . 12 𝐵 = ∅ → ∃𝑦 𝑦𝐵)
3938adantr 484 . . . . . . . . . . 11 ((¬ 𝐵 = ∅ ∧ (𝐹𝑥) = ∅) → ∃𝑦 𝑦𝐵)
40 id 22 . . . . . . . . . . . . . . . 16 ((𝐹𝑥) = ∅ → (𝐹𝑥) = ∅)
41 0ss 4295 . . . . . . . . . . . . . . . 16 ∅ ⊆ 𝑦
4240, 41eqsstrdi 3941 . . . . . . . . . . . . . . 15 ((𝐹𝑥) = ∅ → (𝐹𝑥) ⊆ 𝑦)
4342anim1ci 619 . . . . . . . . . . . . . 14 (((𝐹𝑥) = ∅ ∧ 𝑦𝐵) → (𝑦𝐵 ∧ (𝐹𝑥) ⊆ 𝑦))
4443ex 416 . . . . . . . . . . . . 13 ((𝐹𝑥) = ∅ → (𝑦𝐵 → (𝑦𝐵 ∧ (𝐹𝑥) ⊆ 𝑦)))
4544adantl 485 . . . . . . . . . . . 12 ((¬ 𝐵 = ∅ ∧ (𝐹𝑥) = ∅) → (𝑦𝐵 → (𝑦𝐵 ∧ (𝐹𝑥) ⊆ 𝑦)))
4645eximdv 1924 . . . . . . . . . . 11 ((¬ 𝐵 = ∅ ∧ (𝐹𝑥) = ∅) → (∃𝑦 𝑦𝐵 → ∃𝑦(𝑦𝐵 ∧ (𝐹𝑥) ⊆ 𝑦)))
4739, 46mpd 15 . . . . . . . . . 10 ((¬ 𝐵 = ∅ ∧ (𝐹𝑥) = ∅) → ∃𝑦(𝑦𝐵 ∧ (𝐹𝑥) ⊆ 𝑦))
48 df-rex 3060 . . . . . . . . . 10 (∃𝑦𝐵 (𝐹𝑥) ⊆ 𝑦 ↔ ∃𝑦(𝑦𝐵 ∧ (𝐹𝑥) ⊆ 𝑦))
4947, 48sylibr 237 . . . . . . . . 9 ((¬ 𝐵 = ∅ ∧ (𝐹𝑥) = ∅) → ∃𝑦𝐵 (𝐹𝑥) ⊆ 𝑦)
5049ad4ant24 754 . . . . . . . 8 ((((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) ∧ 𝑥𝐴) ∧ (𝐹𝑥) = ∅) → ∃𝑦𝐵 (𝐹𝑥) ⊆ 𝑦)
5136, 50syldan 594 . . . . . . 7 ((((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) ∧ 𝑥𝐴) ∧ ¬ (𝐹𝑥) ∈ 𝐵) → ∃𝑦𝐵 (𝐹𝑥) ⊆ 𝑦)
5229, 51pm2.61dan 813 . . . . . 6 (((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) ∧ 𝑥𝐴) → ∃𝑦𝐵 (𝐹𝑥) ⊆ 𝑦)
5352ralrimiva 3097 . . . . 5 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) → ∀𝑥𝐴𝑦𝐵 (𝐹𝑥) ⊆ 𝑦)
54 iunss2 4945 . . . . 5 (∀𝑥𝐴𝑦𝐵 (𝐹𝑥) ⊆ 𝑦 𝑥𝐴 (𝐹𝑥) ⊆ 𝑦𝐵 𝑦)
5553, 54syl 17 . . . 4 ((𝐹:𝐴onto→(𝐵 ∪ {∅}) ∧ ¬ 𝐵 = ∅) → 𝑥𝐴 (𝐹𝑥) ⊆ 𝑦𝐵 𝑦)
5624, 55pm2.61dan 813 . . 3 (𝐹:𝐴onto→(𝐵 ∪ {∅}) → 𝑥𝐴 (𝐹𝑥) ⊆ 𝑦𝐵 𝑦)
5710, 56eqssd 3904 . 2 (𝐹:𝐴onto→(𝐵 ∪ {∅}) → 𝑦𝐵 𝑦 = 𝑥𝐴 (𝐹𝑥))
581, 57syl5eq 2786 1 (𝐹:𝐴onto→(𝐵 ∪ {∅}) → 𝐵 = 𝑥𝐴 (𝐹𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1542  wex 1786  wcel 2114  wral 3054  wrex 3055  cun 3851  wss 3853  c0 4221  {csn 4526   cuni 4806   ciun 4891  ontowfo 6348  cfv 6350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pr 5306
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ral 3059  df-rex 3060  df-v 3402  df-sbc 3686  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-iun 4893  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-iota 6308  df-fun 6352  df-fn 6353  df-f 6354  df-fo 6356  df-fv 6358
This theorem is referenced by:  ismeannd  43588
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