Theorem founiiun 42715

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:𝐴–onto→𝐵 → ∪ 𝐵 = ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹

Proof of Theorem founiiun

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		uniiun 4988	. 2 ⊢ ∪ 𝐵 = ∪ 𝑦 ∈ 𝐵 𝑦
2		foelrni 6831	. . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦)
3		eqimss2 3978	. . . . . . 7 ⊢ ((𝐹‘𝑥) = 𝑦 → 𝑦 ⊆ (𝐹‘𝑥))
4	3	reximi 3178	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦 → ∃𝑥 ∈ 𝐴 𝑦 ⊆ (𝐹‘𝑥))
5	2, 4	syl 17	. . . . 5 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 ⊆ (𝐹‘𝑥))
6	5	ralrimiva 3103	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 ⊆ (𝐹‘𝑥))
7		iunss2 4979	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 ⊆ (𝐹‘𝑥) → ∪ 𝑦 ∈ 𝐵 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥))
8	6, 7	syl 17	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → ∪ 𝑦 ∈ 𝐵 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥))
9		fof 6688	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
10	9	ffvelrnda 6961	. . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
11		ssidd 3944	. . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ⊆ (𝐹‘𝑥))
12		sseq2 3947	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑥) → ((𝐹‘𝑥) ⊆ 𝑦 ↔ (𝐹‘𝑥) ⊆ (𝐹‘𝑥)))
13	12	rspcev 3561	. . . . . 6 ⊢ (((𝐹‘𝑥) ∈ 𝐵 ∧ (𝐹‘𝑥) ⊆ (𝐹‘𝑥)) → ∃𝑦 ∈ 𝐵 (𝐹‘𝑥) ⊆ 𝑦)
14	10, 11, 13	syl2anc 584	. . . . 5 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 (𝐹‘𝑥) ⊆ 𝑦)
15	14	ralrimiva 3103	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐹‘𝑥) ⊆ 𝑦)
16		iunss2 4979	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐹‘𝑥) ⊆ 𝑦 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) ⊆ ∪ 𝑦 ∈ 𝐵 𝑦)
17	15, 16	syl 17	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) ⊆ ∪ 𝑦 ∈ 𝐵 𝑦)
18	8, 17	eqssd 3938	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → ∪ 𝑦 ∈ 𝐵 𝑦 = ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥))
19	1, 18	eqtrid 2790	1 ⊢ (𝐹:𝐴–onto→𝐵 → ∪ 𝐵 = ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065   ⊆ wss 3887  ∪ cuni 4839  ∪ ciun 4924  –onto→wfo 6431  ‘cfv 6433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fo 6439  df-fv 6441

This theorem is referenced by:  founiiun0  42728  issalnnd  43884  caragenunicl  44062  isomenndlem  44068