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Theorem founiiun 43644
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 5054 . 2 𝐵 = 𝑦𝐵 𝑦
2 foelcdmi 6940 . . . . . 6 ((𝐹:𝐴onto𝐵𝑦𝐵) → ∃𝑥𝐴 (𝐹𝑥) = 𝑦)
3 eqimss2 4037 . . . . . . 7 ((𝐹𝑥) = 𝑦𝑦 ⊆ (𝐹𝑥))
43reximi 3083 . . . . . 6 (∃𝑥𝐴 (𝐹𝑥) = 𝑦 → ∃𝑥𝐴 𝑦 ⊆ (𝐹𝑥))
52, 4syl 17 . . . . 5 ((𝐹:𝐴onto𝐵𝑦𝐵) → ∃𝑥𝐴 𝑦 ⊆ (𝐹𝑥))
65ralrimiva 3145 . . . 4 (𝐹:𝐴onto𝐵 → ∀𝑦𝐵𝑥𝐴 𝑦 ⊆ (𝐹𝑥))
7 iunss2 5045 . . . 4 (∀𝑦𝐵𝑥𝐴 𝑦 ⊆ (𝐹𝑥) → 𝑦𝐵 𝑦 𝑥𝐴 (𝐹𝑥))
86, 7syl 17 . . 3 (𝐹:𝐴onto𝐵 𝑦𝐵 𝑦 𝑥𝐴 (𝐹𝑥))
9 fof 6792 . . . . . . 7 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
109ffvelcdmda 7071 . . . . . 6 ((𝐹:𝐴onto𝐵𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
11 ssidd 4001 . . . . . 6 ((𝐹:𝐴onto𝐵𝑥𝐴) → (𝐹𝑥) ⊆ (𝐹𝑥))
12 sseq2 4004 . . . . . . 7 (𝑦 = (𝐹𝑥) → ((𝐹𝑥) ⊆ 𝑦 ↔ (𝐹𝑥) ⊆ (𝐹𝑥)))
1312rspcev 3609 . . . . . 6 (((𝐹𝑥) ∈ 𝐵 ∧ (𝐹𝑥) ⊆ (𝐹𝑥)) → ∃𝑦𝐵 (𝐹𝑥) ⊆ 𝑦)
1410, 11, 13syl2anc 584 . . . . 5 ((𝐹:𝐴onto𝐵𝑥𝐴) → ∃𝑦𝐵 (𝐹𝑥) ⊆ 𝑦)
1514ralrimiva 3145 . . . 4 (𝐹:𝐴onto𝐵 → ∀𝑥𝐴𝑦𝐵 (𝐹𝑥) ⊆ 𝑦)
16 iunss2 5045 . . . 4 (∀𝑥𝐴𝑦𝐵 (𝐹𝑥) ⊆ 𝑦 𝑥𝐴 (𝐹𝑥) ⊆ 𝑦𝐵 𝑦)
1715, 16syl 17 . . 3 (𝐹:𝐴onto𝐵 𝑥𝐴 (𝐹𝑥) ⊆ 𝑦𝐵 𝑦)
188, 17eqssd 3995 . 2 (𝐹:𝐴onto𝐵 𝑦𝐵 𝑦 = 𝑥𝐴 (𝐹𝑥))
191, 18eqtrid 2783 1 (𝐹:𝐴onto𝐵 𝐵 = 𝑥𝐴 (𝐹𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3060  wrex 3069  wss 3944   cuni 4901   ciun 4990  ontowfo 6530  cfv 6532
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-fo 6538  df-fv 6540
This theorem is referenced by:  founiiun0  43657  issalnnd  44832  caragenunicl  45011  isomenndlem  45017
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