Theorem metakunt17 41001

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 513	. . . . . 6 ⊢ (𝜑 → ((𝐴 ∩ 𝐵) = ∅ ∧ (𝑋 ∩ 𝑌) = ∅))
6	1, 2, 5	jca31 516	. . . . 5 ⊢ (𝜑 → ((𝐺:𝐴–1-1-onto→𝑋 ∧ 𝐻:𝐵–1-1-onto→𝑌) ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ (𝑋 ∩ 𝑌) = ∅)))
7		f1oun 6853	. . . . 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 4276	. . . . . 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 4165	. . . . . . 7 ⊢ (𝜑 → ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ 𝐶)) = (∅ ∪ ∅))
14		0un 4393	. . . . . . . 8 ⊢ (∅ ∪ ∅) = ∅
15	14	a1i 11	. . . . . . 7 ⊢ (𝜑 → (∅ ∪ ∅) = ∅)
16	13, 15	eqtrd 2773	. . . . . 6 ⊢ (𝜑 → ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ 𝐶)) = ∅)
17	10, 16	eqtrid 2785	. . . . 5 ⊢ (𝜑 → ((𝐴 ∪ 𝐵) ∩ 𝐶) = ∅)
18		indir 4276	. . . . . 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 4165	. . . . . . 7 ⊢ (𝜑 → ((𝑋 ∩ 𝑍) ∪ (𝑌 ∩ 𝑍)) = (∅ ∪ ∅))
22	21, 15	eqtrd 2773	. . . . . 6 ⊢ (𝜑 → ((𝑋 ∩ 𝑍) ∪ (𝑌 ∩ 𝑍)) = ∅)
23	18, 22	eqtrid 2785	. . . . 5 ⊢ (𝜑 → ((𝑋 ∪ 𝑌) ∩ 𝑍) = ∅)
24	17, 23	jca 513	. . . 4 ⊢ (𝜑 → (((𝐴 ∪ 𝐵) ∩ 𝐶) = ∅ ∧ ((𝑋 ∪ 𝑌) ∩ 𝑍) = ∅))
25	8, 9, 24	jca31 516	. . 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 6853	. . 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 6828	. 2 ⊢ (𝜑 → (𝐹:𝐷–1-1-onto→𝑊 ↔ ((𝐺 ∪ 𝐻) ∪ 𝐼):((𝐴 ∪ 𝐵) ∪ 𝐶)–1-1-onto→((𝑋 ∪ 𝑌) ∪ 𝑍)))
32	27, 31	mpbird 257	1 ⊢ (𝜑 → 𝐹:𝐷–1-1-onto→𝑊)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∪ cun 3947   ∩ cin 3948  ∅c0 4323  –1-1-onto→wf1o 6543

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551

This theorem is referenced by:  metakunt25  41009