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Theorem founiiun 39880
 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 4707 . . 3 𝐵 = 𝑦𝐵 𝑦
21a1i 11 . 2 (𝐹:𝐴onto𝐵 𝐵 = 𝑦𝐵 𝑦)
3 simpl 468 . . . . . . 7 ((𝐹:𝐴onto𝐵𝑦𝐵) → 𝐹:𝐴onto𝐵)
4 simpr 471 . . . . . . 7 ((𝐹:𝐴onto𝐵𝑦𝐵) → 𝑦𝐵)
5 foelrni 6386 . . . . . . 7 ((𝐹:𝐴onto𝐵𝑦𝐵) → ∃𝑥𝐴 (𝐹𝑥) = 𝑦)
63, 4, 5syl2anc 573 . . . . . 6 ((𝐹:𝐴onto𝐵𝑦𝐵) → ∃𝑥𝐴 (𝐹𝑥) = 𝑦)
7 eqimss2 3807 . . . . . . . 8 ((𝐹𝑥) = 𝑦𝑦 ⊆ (𝐹𝑥))
87reximi 3159 . . . . . . 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 3115 . . . 4 (𝐹:𝐴onto𝐵 → ∀𝑦𝐵𝑥𝐴 𝑦 ⊆ (𝐹𝑥))
12 iunss2 4699 . . . 4 (∀𝑦𝐵𝑥𝐴 𝑦 ⊆ (𝐹𝑥) → 𝑦𝐵 𝑦 𝑥𝐴 (𝐹𝑥))
1311, 12syl 17 . . 3 (𝐹:𝐴onto𝐵 𝑦𝐵 𝑦 𝑥𝐴 (𝐹𝑥))
14 fof 6256 . . . . . . 7 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
1514ffvelrnda 6502 . . . . . 6 ((𝐹:𝐴onto𝐵𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
16 ssid 3773 . . . . . . 7 (𝐹𝑥) ⊆ (𝐹𝑥)
1716a1i 11 . . . . . 6 ((𝐹:𝐴onto𝐵𝑥𝐴) → (𝐹𝑥) ⊆ (𝐹𝑥))
18 sseq2 3776 . . . . . . 7 (𝑦 = (𝐹𝑥) → ((𝐹𝑥) ⊆ 𝑦 ↔ (𝐹𝑥) ⊆ (𝐹𝑥)))
1918rspcev 3460 . . . . . 6 (((𝐹𝑥) ∈ 𝐵 ∧ (𝐹𝑥) ⊆ (𝐹𝑥)) → ∃𝑦𝐵 (𝐹𝑥) ⊆ 𝑦)
2015, 17, 19syl2anc 573 . . . . 5 ((𝐹:𝐴onto𝐵𝑥𝐴) → ∃𝑦𝐵 (𝐹𝑥) ⊆ 𝑦)
2120ralrimiva 3115 . . . 4 (𝐹:𝐴onto𝐵 → ∀𝑥𝐴𝑦𝐵 (𝐹𝑥) ⊆ 𝑦)
22 iunss2 4699 . . . 4 (∀𝑥𝐴𝑦𝐵 (𝐹𝑥) ⊆ 𝑦 𝑥𝐴 (𝐹𝑥) ⊆ 𝑦𝐵 𝑦)
2321, 22syl 17 . . 3 (𝐹:𝐴onto𝐵 𝑥𝐴 (𝐹𝑥) ⊆ 𝑦𝐵 𝑦)
2413, 23eqssd 3769 . 2 (𝐹:𝐴onto𝐵 𝑦𝐵 𝑦 = 𝑥𝐴 (𝐹𝑥))
252, 24eqtrd 2805 1 (𝐹:𝐴onto𝐵 𝐵 = 𝑥𝐴 (𝐹𝑥))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   = wceq 1631   ∈ wcel 2145  ∀wral 3061  ∃wrex 3062   ⊆ wss 3723  ∪ cuni 4574  ∪ ciun 4654  –onto→wfo 6029  ‘cfv 6031 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fo 6037  df-fv 6039 This theorem is referenced by:  founiiun0  39897  issalnnd  41080  caragenunicl  41258  isomenndlem  41264
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