Theorem metakunt17 41307

Description: The union of three disjoint bijections is a bijection. (Contributed by metakunt, 28-May-2024.)

Hypotheses

Ref	Expression
metakunt17.1	⊢ (𝜑 → 𝐺:𝐴–1-1-onto→𝑋)
metakunt17.2	⊢ (𝜑 → 𝐻:𝐵–1-1-onto→𝑌)
metakunt17.3	⊢ (𝜑 → 𝐼:𝐶–1-1-onto→𝑍)
metakunt17.4	⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)
metakunt17.5	⊢ (𝜑 → (𝐴 ∩ 𝐶) = ∅)
metakunt17.6	⊢ (𝜑 → (𝐵 ∩ 𝐶) = ∅)
metakunt17.7	⊢ (𝜑 → (𝑋 ∩ 𝑌) = ∅)
metakunt17.8	⊢ (𝜑 → (𝑋 ∩ 𝑍) = ∅)
metakunt17.9	⊢ (𝜑 → (𝑌 ∩ 𝑍) = ∅)
metakunt17.10	⊢ (𝜑 → 𝐹 = ((𝐺 ∪ 𝐻) ∪ 𝐼))
metakunt17.11	⊢ (𝜑 → 𝐷 = ((𝐴 ∪ 𝐵) ∪ 𝐶))
metakunt17.12	⊢ (𝜑 → 𝑊 = ((𝑋 ∪ 𝑌) ∪ 𝑍))

Assertion

Ref	Expression
metakunt17	⊢ (𝜑 → 𝐹:𝐷–1-1-onto→𝑊)

Proof of Theorem metakunt17

Step	Hyp	Ref	Expression
1		metakunt17.1	. . . . . 6 ⊢ (𝜑 → 𝐺:𝐴–1-1-onto→𝑋)
2		metakunt17.2	. . . . . 6 ⊢ (𝜑 → 𝐻:𝐵–1-1-onto→𝑌)
3		metakunt17.4	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)
4		metakunt17.7	. . . . . . 7 ⊢ (𝜑 → (𝑋 ∩ 𝑌) = ∅)
5	3, 4	jca 510	. . . . . 6 ⊢ (𝜑 → ((𝐴 ∩ 𝐵) = ∅ ∧ (𝑋 ∩ 𝑌) = ∅))
6	1, 2, 5	jca31 513	. . . . 5 ⊢ (𝜑 → ((𝐺:𝐴–1-1-onto→𝑋 ∧ 𝐻:𝐵–1-1-onto→𝑌) ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ (𝑋 ∩ 𝑌) = ∅)))
7		f1oun 6851	. . . . 5 ⊢ (((𝐺:𝐴–1-1-onto→𝑋 ∧ 𝐻:𝐵–1-1-onto→𝑌) ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ (𝑋 ∩ 𝑌) = ∅)) → (𝐺 ∪ 𝐻):(𝐴 ∪ 𝐵)–1-1-onto→(𝑋 ∪ 𝑌))
8	6, 7	syl 17	. . . 4 ⊢ (𝜑 → (𝐺 ∪ 𝐻):(𝐴 ∪ 𝐵)–1-1-onto→(𝑋 ∪ 𝑌))
9		metakunt17.3	. . . 4 ⊢ (𝜑 → 𝐼:𝐶–1-1-onto→𝑍)
10		indir 4274	. . . . . 6 ⊢ ((𝐴 ∪ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ 𝐶))
11		metakunt17.5	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∩ 𝐶) = ∅)
12		metakunt17.6	. . . . . . . 8 ⊢ (𝜑 → (𝐵 ∩ 𝐶) = ∅)
13	11, 12	uneq12d 4163	. . . . . . 7 ⊢ (𝜑 → ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ 𝐶)) = (∅ ∪ ∅))
14		0un 4391	. . . . . . . 8 ⊢ (∅ ∪ ∅) = ∅
15	14	a1i 11	. . . . . . 7 ⊢ (𝜑 → (∅ ∪ ∅) = ∅)
16	13, 15	eqtrd 2770	. . . . . 6 ⊢ (𝜑 → ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ 𝐶)) = ∅)
17	10, 16	eqtrid 2782	. . . . 5 ⊢ (𝜑 → ((𝐴 ∪ 𝐵) ∩ 𝐶) = ∅)
18		indir 4274	. . . . . 6 ⊢ ((𝑋 ∪ 𝑌) ∩ 𝑍) = ((𝑋 ∩ 𝑍) ∪ (𝑌 ∩ 𝑍))
19		metakunt17.8	. . . . . . . 8 ⊢ (𝜑 → (𝑋 ∩ 𝑍) = ∅)
20		metakunt17.9	. . . . . . . 8 ⊢ (𝜑 → (𝑌 ∩ 𝑍) = ∅)
21	19, 20	uneq12d 4163	. . . . . . 7 ⊢ (𝜑 → ((𝑋 ∩ 𝑍) ∪ (𝑌 ∩ 𝑍)) = (∅ ∪ ∅))
22	21, 15	eqtrd 2770	. . . . . 6 ⊢ (𝜑 → ((𝑋 ∩ 𝑍) ∪ (𝑌 ∩ 𝑍)) = ∅)
23	18, 22	eqtrid 2782	. . . . 5 ⊢ (𝜑 → ((𝑋 ∪ 𝑌) ∩ 𝑍) = ∅)
24	17, 23	jca 510	. . . 4 ⊢ (𝜑 → (((𝐴 ∪ 𝐵) ∩ 𝐶) = ∅ ∧ ((𝑋 ∪ 𝑌) ∩ 𝑍) = ∅))
25	8, 9, 24	jca31 513	. . 3 ⊢ (𝜑 → (((𝐺 ∪ 𝐻):(𝐴 ∪ 𝐵)–1-1-onto→(𝑋 ∪ 𝑌) ∧ 𝐼:𝐶–1-1-onto→𝑍) ∧ (((𝐴 ∪ 𝐵) ∩ 𝐶) = ∅ ∧ ((𝑋 ∪ 𝑌) ∩ 𝑍) = ∅)))
26		f1oun 6851	. . 3 ⊢ ((((𝐺 ∪ 𝐻):(𝐴 ∪ 𝐵)–1-1-onto→(𝑋 ∪ 𝑌) ∧ 𝐼:𝐶–1-1-onto→𝑍) ∧ (((𝐴 ∪ 𝐵) ∩ 𝐶) = ∅ ∧ ((𝑋 ∪ 𝑌) ∩ 𝑍) = ∅)) → ((𝐺 ∪ 𝐻) ∪ 𝐼):((𝐴 ∪ 𝐵) ∪ 𝐶)–1-1-onto→((𝑋 ∪ 𝑌) ∪ 𝑍))
27	25, 26	syl 17	. 2 ⊢ (𝜑 → ((𝐺 ∪ 𝐻) ∪ 𝐼):((𝐴 ∪ 𝐵) ∪ 𝐶)–1-1-onto→((𝑋 ∪ 𝑌) ∪ 𝑍))
28		metakunt17.10	. . 3 ⊢ (𝜑 → 𝐹 = ((𝐺 ∪ 𝐻) ∪ 𝐼))
29		metakunt17.11	. . 3 ⊢ (𝜑 → 𝐷 = ((𝐴 ∪ 𝐵) ∪ 𝐶))
30		metakunt17.12	. . 3 ⊢ (𝜑 → 𝑊 = ((𝑋 ∪ 𝑌) ∪ 𝑍))
31	28, 29, 30	f1oeq123d 6826	. 2 ⊢ (𝜑 → (𝐹:𝐷–1-1-onto→𝑊 ↔ ((𝐺 ∪ 𝐻) ∪ 𝐼):((𝐴 ∪ 𝐵) ∪ 𝐶)–1-1-onto→((𝑋 ∪ 𝑌) ∪ 𝑍)))
32	27, 31	mpbird 256	1 ⊢ (𝜑 → 𝐹:𝐷–1-1-onto→𝑊)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1539   ∪ cun 3945   ∩ cin 3946  ∅c0 4321  –1-1-onto→wf1o 6541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549

This theorem is referenced by:  metakunt25  41315