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Theorem founiiun 38831
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 4539 . . 3 𝐵 = 𝑦𝐵 𝑦
21a1i 11 . 2 (𝐹:𝐴onto𝐵 𝐵 = 𝑦𝐵 𝑦)
3 simpl 473 . . . . . . 7 ((𝐹:𝐴onto𝐵𝑦𝐵) → 𝐹:𝐴onto𝐵)
4 simpr 477 . . . . . . 7 ((𝐹:𝐴onto𝐵𝑦𝐵) → 𝑦𝐵)
5 foelrni 6201 . . . . . . 7 ((𝐹:𝐴onto𝐵𝑦𝐵) → ∃𝑥𝐴 (𝐹𝑥) = 𝑦)
63, 4, 5syl2anc 692 . . . . . 6 ((𝐹:𝐴onto𝐵𝑦𝐵) → ∃𝑥𝐴 (𝐹𝑥) = 𝑦)
7 eqimss2 3637 . . . . . . . 8 ((𝐹𝑥) = 𝑦𝑦 ⊆ (𝐹𝑥))
87reximi 3005 . . . . . . 7 (∃𝑥𝐴 (𝐹𝑥) = 𝑦 → ∃𝑥𝐴 𝑦 ⊆ (𝐹𝑥))
98a1i 11 . . . . . 6 ((𝐹:𝐴onto𝐵𝑦𝐵) → (∃𝑥𝐴 (𝐹𝑥) = 𝑦 → ∃𝑥𝐴 𝑦 ⊆ (𝐹𝑥)))
106, 9mpd 15 . . . . 5 ((𝐹:𝐴onto𝐵𝑦𝐵) → ∃𝑥𝐴 𝑦 ⊆ (𝐹𝑥))
1110ralrimiva 2960 . . . 4 (𝐹:𝐴onto𝐵 → ∀𝑦𝐵𝑥𝐴 𝑦 ⊆ (𝐹𝑥))
12 iunss2 4531 . . . 4 (∀𝑦𝐵𝑥𝐴 𝑦 ⊆ (𝐹𝑥) → 𝑦𝐵 𝑦 𝑥𝐴 (𝐹𝑥))
1311, 12syl 17 . . 3 (𝐹:𝐴onto𝐵 𝑦𝐵 𝑦 𝑥𝐴 (𝐹𝑥))
14 fof 6072 . . . . . . 7 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
1514ffvelrnda 6315 . . . . . 6 ((𝐹:𝐴onto𝐵𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
16 ssid 3603 . . . . . . 7 (𝐹𝑥) ⊆ (𝐹𝑥)
1716a1i 11 . . . . . 6 ((𝐹:𝐴onto𝐵𝑥𝐴) → (𝐹𝑥) ⊆ (𝐹𝑥))
18 sseq2 3606 . . . . . . 7 (𝑦 = (𝐹𝑥) → ((𝐹𝑥) ⊆ 𝑦 ↔ (𝐹𝑥) ⊆ (𝐹𝑥)))
1918rspcev 3295 . . . . . 6 (((𝐹𝑥) ∈ 𝐵 ∧ (𝐹𝑥) ⊆ (𝐹𝑥)) → ∃𝑦𝐵 (𝐹𝑥) ⊆ 𝑦)
2015, 17, 19syl2anc 692 . . . . 5 ((𝐹:𝐴onto𝐵𝑥𝐴) → ∃𝑦𝐵 (𝐹𝑥) ⊆ 𝑦)
2120ralrimiva 2960 . . . 4 (𝐹:𝐴onto𝐵 → ∀𝑥𝐴𝑦𝐵 (𝐹𝑥) ⊆ 𝑦)
22 iunss2 4531 . . . 4 (∀𝑥𝐴𝑦𝐵 (𝐹𝑥) ⊆ 𝑦 𝑥𝐴 (𝐹𝑥) ⊆ 𝑦𝐵 𝑦)
2321, 22syl 17 . . 3 (𝐹:𝐴onto𝐵 𝑥𝐴 (𝐹𝑥) ⊆ 𝑦𝐵 𝑦)
2413, 23eqssd 3600 . 2 (𝐹:𝐴onto𝐵 𝑦𝐵 𝑦 = 𝑥𝐴 (𝐹𝑥))
252, 24eqtrd 2655 1 (𝐹:𝐴onto𝐵 𝐵 = 𝑥𝐴 (𝐹𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  wss 3555   cuni 4402   ciun 4485  ontowfo 5845  cfv 5847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fo 5853  df-fv 5855
This theorem is referenced by:  founiiun0  38848  issalnnd  39867  caragenunicl  40042  isomenndlem  40048
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