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: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 4707	. . 3 ⊢ ∪ 𝐵 = ∪ 𝑦 ∈ 𝐵 𝑦
2	1	a1i 11	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → ∪ 𝐵 = ∪ 𝑦 ∈ 𝐵 𝑦)
3		simpl 468	. . . . . . 7 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑦 ∈ 𝐵) → 𝐹:𝐴–onto→𝐵)
4		simpr 471	. . . . . . 7 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
5		foelrni 6386	. . . . . . 7 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦)
6	3, 4, 5	syl2anc 573	. . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦)
7		eqimss2 3807	. . . . . . . 8 ⊢ ((𝐹‘𝑥) = 𝑦 → 𝑦 ⊆ (𝐹‘𝑥))
8	7	reximi 3159	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦 → ∃𝑥 ∈ 𝐴 𝑦 ⊆ (𝐹‘𝑥))
9	8	a1i 11	. . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑦 ∈ 𝐵) → (∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦 → ∃𝑥 ∈ 𝐴 𝑦 ⊆ (𝐹‘𝑥)))
10	6, 9	mpd 15	. . . . 5 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 ⊆ (𝐹‘𝑥))
11	10	ralrimiva 3115	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 ⊆ (𝐹‘𝑥))
12		iunss2 4699	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 ⊆ (𝐹‘𝑥) → ∪ 𝑦 ∈ 𝐵 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥))
13	11, 12	syl 17	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → ∪ 𝑦 ∈ 𝐵 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥))
14		fof 6256	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
15	14	ffvelrnda 6502	. . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
16		ssid 3773	. . . . . . 7 ⊢ (𝐹‘𝑥) ⊆ (𝐹‘𝑥)
17	16	a1i 11	. . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ⊆ (𝐹‘𝑥))
18		sseq2 3776	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑥) → ((𝐹‘𝑥) ⊆ 𝑦 ↔ (𝐹‘𝑥) ⊆ (𝐹‘𝑥)))
19	18	rspcev 3460	. . . . . 6 ⊢ (((𝐹‘𝑥) ∈ 𝐵 ∧ (𝐹‘𝑥) ⊆ (𝐹‘𝑥)) → ∃𝑦 ∈ 𝐵 (𝐹‘𝑥) ⊆ 𝑦)
20	15, 17, 19	syl2anc 573	. . . . 5 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 (𝐹‘𝑥) ⊆ 𝑦)
21	20	ralrimiva 3115	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐹‘𝑥) ⊆ 𝑦)
22		iunss2 4699	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐹‘𝑥) ⊆ 𝑦 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) ⊆ ∪ 𝑦 ∈ 𝐵 𝑦)
23	21, 22	syl 17	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) ⊆ ∪ 𝑦 ∈ 𝐵 𝑦)
24	13, 23	eqssd 3769	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → ∪ 𝑦 ∈ 𝐵 𝑦 = ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥))
25	2, 24	eqtrd 2805	1 ⊢ (𝐹:𝐴–onto→𝐵 → ∪ 𝐵 = ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥))

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