Metamath Proof Explorer


Theorem metakunt17

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

Ref Expression
Hypotheses metakunt17.1 ( 𝜑𝐺 : 𝐴1-1-onto𝑋 )
metakunt17.2 ( 𝜑𝐻 : 𝐵1-1-onto𝑌 )
metakunt17.3 ( 𝜑𝐼 : 𝐶1-1-onto𝑍 )
metakunt17.4 ( 𝜑 → ( 𝐴𝐵 ) = ∅ )
metakunt17.5 ( 𝜑 → ( 𝐴𝐶 ) = ∅ )
metakunt17.6 ( 𝜑 → ( 𝐵𝐶 ) = ∅ )
metakunt17.7 ( 𝜑 → ( 𝑋𝑌 ) = ∅ )
metakunt17.8 ( 𝜑 → ( 𝑋𝑍 ) = ∅ )
metakunt17.9 ( 𝜑 → ( 𝑌𝑍 ) = ∅ )
metakunt17.10 ( 𝜑𝐹 = ( ( 𝐺𝐻 ) ∪ 𝐼 ) )
metakunt17.11 ( 𝜑𝐷 = ( ( 𝐴𝐵 ) ∪ 𝐶 ) )
metakunt17.12 ( 𝜑𝑊 = ( ( 𝑋𝑌 ) ∪ 𝑍 ) )
Assertion metakunt17 ( 𝜑𝐹 : 𝐷1-1-onto𝑊 )

Proof

Step Hyp Ref Expression
1 metakunt17.1 ( 𝜑𝐺 : 𝐴1-1-onto𝑋 )
2 metakunt17.2 ( 𝜑𝐻 : 𝐵1-1-onto𝑌 )
3 metakunt17.3 ( 𝜑𝐼 : 𝐶1-1-onto𝑍 )
4 metakunt17.4 ( 𝜑 → ( 𝐴𝐵 ) = ∅ )
5 metakunt17.5 ( 𝜑 → ( 𝐴𝐶 ) = ∅ )
6 metakunt17.6 ( 𝜑 → ( 𝐵𝐶 ) = ∅ )
7 metakunt17.7 ( 𝜑 → ( 𝑋𝑌 ) = ∅ )
8 metakunt17.8 ( 𝜑 → ( 𝑋𝑍 ) = ∅ )
9 metakunt17.9 ( 𝜑 → ( 𝑌𝑍 ) = ∅ )
10 metakunt17.10 ( 𝜑𝐹 = ( ( 𝐺𝐻 ) ∪ 𝐼 ) )
11 metakunt17.11 ( 𝜑𝐷 = ( ( 𝐴𝐵 ) ∪ 𝐶 ) )
12 metakunt17.12 ( 𝜑𝑊 = ( ( 𝑋𝑌 ) ∪ 𝑍 ) )
13 4 7 jca ( 𝜑 → ( ( 𝐴𝐵 ) = ∅ ∧ ( 𝑋𝑌 ) = ∅ ) )
14 1 2 13 jca31 ( 𝜑 → ( ( 𝐺 : 𝐴1-1-onto𝑋𝐻 : 𝐵1-1-onto𝑌 ) ∧ ( ( 𝐴𝐵 ) = ∅ ∧ ( 𝑋𝑌 ) = ∅ ) ) )
15 f1oun ( ( ( 𝐺 : 𝐴1-1-onto𝑋𝐻 : 𝐵1-1-onto𝑌 ) ∧ ( ( 𝐴𝐵 ) = ∅ ∧ ( 𝑋𝑌 ) = ∅ ) ) → ( 𝐺𝐻 ) : ( 𝐴𝐵 ) –1-1-onto→ ( 𝑋𝑌 ) )
16 14 15 syl ( 𝜑 → ( 𝐺𝐻 ) : ( 𝐴𝐵 ) –1-1-onto→ ( 𝑋𝑌 ) )
17 indir ( ( 𝐴𝐵 ) ∩ 𝐶 ) = ( ( 𝐴𝐶 ) ∪ ( 𝐵𝐶 ) )
18 5 6 uneq12d ( 𝜑 → ( ( 𝐴𝐶 ) ∪ ( 𝐵𝐶 ) ) = ( ∅ ∪ ∅ ) )
19 0un ( ∅ ∪ ∅ ) = ∅
20 19 a1i ( 𝜑 → ( ∅ ∪ ∅ ) = ∅ )
21 18 20 eqtrd ( 𝜑 → ( ( 𝐴𝐶 ) ∪ ( 𝐵𝐶 ) ) = ∅ )
22 17 21 syl5eq ( 𝜑 → ( ( 𝐴𝐵 ) ∩ 𝐶 ) = ∅ )
23 indir ( ( 𝑋𝑌 ) ∩ 𝑍 ) = ( ( 𝑋𝑍 ) ∪ ( 𝑌𝑍 ) )
24 8 9 uneq12d ( 𝜑 → ( ( 𝑋𝑍 ) ∪ ( 𝑌𝑍 ) ) = ( ∅ ∪ ∅ ) )
25 24 20 eqtrd ( 𝜑 → ( ( 𝑋𝑍 ) ∪ ( 𝑌𝑍 ) ) = ∅ )
26 23 25 syl5eq ( 𝜑 → ( ( 𝑋𝑌 ) ∩ 𝑍 ) = ∅ )
27 22 26 jca ( 𝜑 → ( ( ( 𝐴𝐵 ) ∩ 𝐶 ) = ∅ ∧ ( ( 𝑋𝑌 ) ∩ 𝑍 ) = ∅ ) )
28 16 3 27 jca31 ( 𝜑 → ( ( ( 𝐺𝐻 ) : ( 𝐴𝐵 ) –1-1-onto→ ( 𝑋𝑌 ) ∧ 𝐼 : 𝐶1-1-onto𝑍 ) ∧ ( ( ( 𝐴𝐵 ) ∩ 𝐶 ) = ∅ ∧ ( ( 𝑋𝑌 ) ∩ 𝑍 ) = ∅ ) ) )
29 f1oun ( ( ( ( 𝐺𝐻 ) : ( 𝐴𝐵 ) –1-1-onto→ ( 𝑋𝑌 ) ∧ 𝐼 : 𝐶1-1-onto𝑍 ) ∧ ( ( ( 𝐴𝐵 ) ∩ 𝐶 ) = ∅ ∧ ( ( 𝑋𝑌 ) ∩ 𝑍 ) = ∅ ) ) → ( ( 𝐺𝐻 ) ∪ 𝐼 ) : ( ( 𝐴𝐵 ) ∪ 𝐶 ) –1-1-onto→ ( ( 𝑋𝑌 ) ∪ 𝑍 ) )
30 28 29 syl ( 𝜑 → ( ( 𝐺𝐻 ) ∪ 𝐼 ) : ( ( 𝐴𝐵 ) ∪ 𝐶 ) –1-1-onto→ ( ( 𝑋𝑌 ) ∪ 𝑍 ) )
31 10 11 12 f1oeq123d ( 𝜑 → ( 𝐹 : 𝐷1-1-onto𝑊 ↔ ( ( 𝐺𝐻 ) ∪ 𝐼 ) : ( ( 𝐴𝐵 ) ∪ 𝐶 ) –1-1-onto→ ( ( 𝑋𝑌 ) ∪ 𝑍 ) ) )
32 30 31 mpbird ( 𝜑𝐹 : 𝐷1-1-onto𝑊 )