Theorem founiiun 42604

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 4984	. 2 ⊢ ∪ 𝐵 = ∪ 𝑦 ∈ 𝐵 𝑦
2		foelrni 6813	. . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦)
3		eqimss2 3974	. . . . . . 7 ⊢ ((𝐹‘𝑥) = 𝑦 → 𝑦 ⊆ (𝐹‘𝑥))
4	3	reximi 3174	. . . . . 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 3107	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 ⊆ (𝐹‘𝑥))
7		iunss2 4975	. . . 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 6672	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
10	9	ffvelrnda 6943	. . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
11		ssidd 3940	. . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ⊆ (𝐹‘𝑥))
12		sseq2 3943	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑥) → ((𝐹‘𝑥) ⊆ 𝑦 ↔ (𝐹‘𝑥) ⊆ (𝐹‘𝑥)))
13	12	rspcev 3552	. . . . . 6 ⊢ (((𝐹‘𝑥) ∈ 𝐵 ∧ (𝐹‘𝑥) ⊆ (𝐹‘𝑥)) → ∃𝑦 ∈ 𝐵 (𝐹‘𝑥) ⊆ 𝑦)
14	10, 11, 13	syl2anc 583	. . . . 5 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 (𝐹‘𝑥) ⊆ 𝑦)
15	14	ralrimiva 3107	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐹‘𝑥) ⊆ 𝑦)
16		iunss2 4975	. . . 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 3934	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → ∪ 𝑦 ∈ 𝐵 𝑦 = ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥))
19	1, 18	syl5eq 2791	1 ⊢ (𝐹:𝐴–onto→𝐵 → ∪ 𝐵 = ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064   ⊆ wss 3883  ∪ cuni 4836  ∪ ciun 4921  –onto→wfo 6416  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fo 6424  df-fv 6426

This theorem is referenced by:  founiiun0  42617  issalnnd  43774  caragenunicl  43952  isomenndlem  43958