Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  founiiun Structured version   Visualization version   GIF version

Theorem founiiun 42209
 Description: Union expressed as an indexed union, when a map onto is given. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Assertion
Ref Expression
founiiun (𝐹:𝐴onto𝐵 𝐵 = 𝑥𝐴 (𝐹𝑥))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹

Proof of Theorem founiiun
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 uniiun 4950 . 2 𝐵 = 𝑦𝐵 𝑦
2 foelrni 6720 . . . . . 6 ((𝐹:𝐴onto𝐵𝑦𝐵) → ∃𝑥𝐴 (𝐹𝑥) = 𝑦)
3 eqimss2 3951 . . . . . . 7 ((𝐹𝑥) = 𝑦𝑦 ⊆ (𝐹𝑥))
43reximi 3171 . . . . . 6 (∃𝑥𝐴 (𝐹𝑥) = 𝑦 → ∃𝑥𝐴 𝑦 ⊆ (𝐹𝑥))
52, 4syl 17 . . . . 5 ((𝐹:𝐴onto𝐵𝑦𝐵) → ∃𝑥𝐴 𝑦 ⊆ (𝐹𝑥))
65ralrimiva 3113 . . . 4 (𝐹:𝐴onto𝐵 → ∀𝑦𝐵𝑥𝐴 𝑦 ⊆ (𝐹𝑥))
7 iunss2 4941 . . . 4 (∀𝑦𝐵𝑥𝐴 𝑦 ⊆ (𝐹𝑥) → 𝑦𝐵 𝑦 𝑥𝐴 (𝐹𝑥))
86, 7syl 17 . . 3 (𝐹:𝐴onto𝐵 𝑦𝐵 𝑦 𝑥𝐴 (𝐹𝑥))
9 fof 6581 . . . . . . 7 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
109ffvelrnda 6848 . . . . . 6 ((𝐹:𝐴onto𝐵𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
11 ssidd 3917 . . . . . 6 ((𝐹:𝐴onto𝐵𝑥𝐴) → (𝐹𝑥) ⊆ (𝐹𝑥))
12 sseq2 3920 . . . . . . 7 (𝑦 = (𝐹𝑥) → ((𝐹𝑥) ⊆ 𝑦 ↔ (𝐹𝑥) ⊆ (𝐹𝑥)))
1312rspcev 3543 . . . . . 6 (((𝐹𝑥) ∈ 𝐵 ∧ (𝐹𝑥) ⊆ (𝐹𝑥)) → ∃𝑦𝐵 (𝐹𝑥) ⊆ 𝑦)
1410, 11, 13syl2anc 587 . . . . 5 ((𝐹:𝐴onto𝐵𝑥𝐴) → ∃𝑦𝐵 (𝐹𝑥) ⊆ 𝑦)
1514ralrimiva 3113 . . . 4 (𝐹:𝐴onto𝐵 → ∀𝑥𝐴𝑦𝐵 (𝐹𝑥) ⊆ 𝑦)
16 iunss2 4941 . . . 4 (∀𝑥𝐴𝑦𝐵 (𝐹𝑥) ⊆ 𝑦 𝑥𝐴 (𝐹𝑥) ⊆ 𝑦𝐵 𝑦)
1715, 16syl 17 . . 3 (𝐹:𝐴onto𝐵 𝑥𝐴 (𝐹𝑥) ⊆ 𝑦𝐵 𝑦)
188, 17eqssd 3911 . 2 (𝐹:𝐴onto𝐵 𝑦𝐵 𝑦 = 𝑥𝐴 (𝐹𝑥))
191, 18syl5eq 2805 1 (𝐹:𝐴onto𝐵 𝐵 = 𝑥𝐴 (𝐹𝑥))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  ∃wrex 3071   ⊆ wss 3860  ∪ cuni 4801  ∪ ciun 4886  –onto→wfo 6338  ‘cfv 6340 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-fo 6346  df-fv 6348 This theorem is referenced by:  founiiun0  42222  issalnnd  43386  caragenunicl  43564  isomenndlem  43570
 Copyright terms: Public domain W3C validator