Theorem funcnvqp

Description: The converse quadruple of ordered pairs is a function if the second members are pairwise different. Note that the second members need not be sets. (Contributed by AV, 23-Jan-2021) (Proof shortened by JJ, 14-Jul-2021)

	Ref	Expression
Assertion	funcnvqp	⊢ ( ( ( ( 𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ) ∧ ( 𝐸 ∈ 𝑊 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( ( 𝐵 ≠ 𝐷 ∧ 𝐵 ≠ 𝐹 ∧ 𝐵 ≠ 𝐻 ) ∧ ( 𝐷 ≠ 𝐹 ∧ 𝐷 ≠ 𝐻 ) ∧ 𝐹 ≠ 𝐻 ) ) → Fun ◡ ( { ⟨ 𝐴 , 𝐵 ⟩ , ⟨ 𝐶 , 𝐷 ⟩ } ∪ { ⟨ 𝐸 , 𝐹 ⟩ , ⟨ 𝐺 , 𝐻 ⟩ } ) )

Proof

Step	Hyp	Ref	Expression
1		funcnvpr	⊢ ( ( 𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷 ) → Fun ◡ { ⟨ 𝐴 , 𝐵 ⟩ , ⟨ 𝐶 , 𝐷 ⟩ } )
2	1	3expa	⊢ ( ( ( 𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ) ∧ 𝐵 ≠ 𝐷 ) → Fun ◡ { ⟨ 𝐴 , 𝐵 ⟩ , ⟨ 𝐶 , 𝐷 ⟩ } )
3	2	3ad2antr1	⊢ ( ( ( 𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ) ∧ ( 𝐵 ≠ 𝐷 ∧ 𝐵 ≠ 𝐹 ∧ 𝐵 ≠ 𝐻 ) ) → Fun ◡ { ⟨ 𝐴 , 𝐵 ⟩ , ⟨ 𝐶 , 𝐷 ⟩ } )
4	3	ad2ant2r	⊢ ( ( ( ( 𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ) ∧ ( 𝐸 ∈ 𝑊 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( ( 𝐵 ≠ 𝐷 ∧ 𝐵 ≠ 𝐹 ∧ 𝐵 ≠ 𝐻 ) ∧ 𝐹 ≠ 𝐻 ) ) → Fun ◡ { ⟨ 𝐴 , 𝐵 ⟩ , ⟨ 𝐶 , 𝐷 ⟩ } )
5	4	3adantr2	⊢ ( ( ( ( 𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ) ∧ ( 𝐸 ∈ 𝑊 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( ( 𝐵 ≠ 𝐷 ∧ 𝐵 ≠ 𝐹 ∧ 𝐵 ≠ 𝐻 ) ∧ ( 𝐷 ≠ 𝐹 ∧ 𝐷 ≠ 𝐻 ) ∧ 𝐹 ≠ 𝐻 ) ) → Fun ◡ { ⟨ 𝐴 , 𝐵 ⟩ , ⟨ 𝐶 , 𝐷 ⟩ } )
6		funcnvpr	⊢ ( ( 𝐸 ∈ 𝑊 ∧ 𝐺 ∈ 𝑇 ∧ 𝐹 ≠ 𝐻 ) → Fun ◡ { ⟨ 𝐸 , 𝐹 ⟩ , ⟨ 𝐺 , 𝐻 ⟩ } )
7	6	3expa	⊢ ( ( ( 𝐸 ∈ 𝑊 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝐹 ≠ 𝐻 ) → Fun ◡ { ⟨ 𝐸 , 𝐹 ⟩ , ⟨ 𝐺 , 𝐻 ⟩ } )
8	7	ad2ant2l	⊢ ( ( ( ( 𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ) ∧ ( 𝐸 ∈ 𝑊 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( ( 𝐵 ≠ 𝐷 ∧ 𝐵 ≠ 𝐹 ∧ 𝐵 ≠ 𝐻 ) ∧ 𝐹 ≠ 𝐻 ) ) → Fun ◡ { ⟨ 𝐸 , 𝐹 ⟩ , ⟨ 𝐺 , 𝐻 ⟩ } )
9	8	3adantr2	⊢ ( ( ( ( 𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ) ∧ ( 𝐸 ∈ 𝑊 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( ( 𝐵 ≠ 𝐷 ∧ 𝐵 ≠ 𝐹 ∧ 𝐵 ≠ 𝐻 ) ∧ ( 𝐷 ≠ 𝐹 ∧ 𝐷 ≠ 𝐻 ) ∧ 𝐹 ≠ 𝐻 ) ) → Fun ◡ { ⟨ 𝐸 , 𝐹 ⟩ , ⟨ 𝐺 , 𝐻 ⟩ } )
10		df-rn	⊢ ran { ⟨ 𝐴 , 𝐵 ⟩ , ⟨ 𝐶 , 𝐷 ⟩ } = dom ◡ { ⟨ 𝐴 , 𝐵 ⟩ , ⟨ 𝐶 , 𝐷 ⟩ }
11		rnpropg	⊢ ( ( 𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ) → ran { ⟨ 𝐴 , 𝐵 ⟩ , ⟨ 𝐶 , 𝐷 ⟩ } = { 𝐵 , 𝐷 } )
12	10 11	eqtr3id	⊢ ( ( 𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ) → dom ◡ { ⟨ 𝐴 , 𝐵 ⟩ , ⟨ 𝐶 , 𝐷 ⟩ } = { 𝐵 , 𝐷 } )
13		df-rn	⊢ ran { ⟨ 𝐸 , 𝐹 ⟩ , ⟨ 𝐺 , 𝐻 ⟩ } = dom ◡ { ⟨ 𝐸 , 𝐹 ⟩ , ⟨ 𝐺 , 𝐻 ⟩ }
14		rnpropg	⊢ ( ( 𝐸 ∈ 𝑊 ∧ 𝐺 ∈ 𝑇 ) → ran { ⟨ 𝐸 , 𝐹 ⟩ , ⟨ 𝐺 , 𝐻 ⟩ } = { 𝐹 , 𝐻 } )
15	13 14	eqtr3id	⊢ ( ( 𝐸 ∈ 𝑊 ∧ 𝐺 ∈ 𝑇 ) → dom ◡ { ⟨ 𝐸 , 𝐹 ⟩ , ⟨ 𝐺 , 𝐻 ⟩ } = { 𝐹 , 𝐻 } )
16	12 15	ineqan12d	⊢ ( ( ( 𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ) ∧ ( 𝐸 ∈ 𝑊 ∧ 𝐺 ∈ 𝑇 ) ) → ( dom ◡ { ⟨ 𝐴 , 𝐵 ⟩ , ⟨ 𝐶 , 𝐷 ⟩ } ∩ dom ◡ { ⟨ 𝐸 , 𝐹 ⟩ , ⟨ 𝐺 , 𝐻 ⟩ } ) = ( { 𝐵 , 𝐷 } ∩ { 𝐹 , 𝐻 } ) )
17		disjpr2	⊢ ( ( ( 𝐵 ≠ 𝐹 ∧ 𝐷 ≠ 𝐹 ) ∧ ( 𝐵 ≠ 𝐻 ∧ 𝐷 ≠ 𝐻 ) ) → ( { 𝐵 , 𝐷 } ∩ { 𝐹 , 𝐻 } ) = ∅ )
18	17	an4s	⊢ ( ( ( 𝐵 ≠ 𝐹 ∧ 𝐵 ≠ 𝐻 ) ∧ ( 𝐷 ≠ 𝐹 ∧ 𝐷 ≠ 𝐻 ) ) → ( { 𝐵 , 𝐷 } ∩ { 𝐹 , 𝐻 } ) = ∅ )
19	18	3adantl1	⊢ ( ( ( 𝐵 ≠ 𝐷 ∧ 𝐵 ≠ 𝐹 ∧ 𝐵 ≠ 𝐻 ) ∧ ( 𝐷 ≠ 𝐹 ∧ 𝐷 ≠ 𝐻 ) ) → ( { 𝐵 , 𝐷 } ∩ { 𝐹 , 𝐻 } ) = ∅ )
20	19	3adant3	⊢ ( ( ( 𝐵 ≠ 𝐷 ∧ 𝐵 ≠ 𝐹 ∧ 𝐵 ≠ 𝐻 ) ∧ ( 𝐷 ≠ 𝐹 ∧ 𝐷 ≠ 𝐻 ) ∧ 𝐹 ≠ 𝐻 ) → ( { 𝐵 , 𝐷 } ∩ { 𝐹 , 𝐻 } ) = ∅ )
21	16 20	sylan9eq	⊢ ( ( ( ( 𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ) ∧ ( 𝐸 ∈ 𝑊 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( ( 𝐵 ≠ 𝐷 ∧ 𝐵 ≠ 𝐹 ∧ 𝐵 ≠ 𝐻 ) ∧ ( 𝐷 ≠ 𝐹 ∧ 𝐷 ≠ 𝐻 ) ∧ 𝐹 ≠ 𝐻 ) ) → ( dom ◡ { ⟨ 𝐴 , 𝐵 ⟩ , ⟨ 𝐶 , 𝐷 ⟩ } ∩ dom ◡ { ⟨ 𝐸 , 𝐹 ⟩ , ⟨ 𝐺 , 𝐻 ⟩ } ) = ∅ )
22		funun	⊢ ( ( ( Fun ◡ { ⟨ 𝐴 , 𝐵 ⟩ , ⟨ 𝐶 , 𝐷 ⟩ } ∧ Fun ◡ { ⟨ 𝐸 , 𝐹 ⟩ , ⟨ 𝐺 , 𝐻 ⟩ } ) ∧ ( dom ◡ { ⟨ 𝐴 , 𝐵 ⟩ , ⟨ 𝐶 , 𝐷 ⟩ } ∩ dom ◡ { ⟨ 𝐸 , 𝐹 ⟩ , ⟨ 𝐺 , 𝐻 ⟩ } ) = ∅ ) → Fun ( ◡ { ⟨ 𝐴 , 𝐵 ⟩ , ⟨ 𝐶 , 𝐷 ⟩ } ∪ ◡ { ⟨ 𝐸 , 𝐹 ⟩ , ⟨ 𝐺 , 𝐻 ⟩ } ) )
23	5 9 21 22	syl21anc	⊢ ( ( ( ( 𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ) ∧ ( 𝐸 ∈ 𝑊 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( ( 𝐵 ≠ 𝐷 ∧ 𝐵 ≠ 𝐹 ∧ 𝐵 ≠ 𝐻 ) ∧ ( 𝐷 ≠ 𝐹 ∧ 𝐷 ≠ 𝐻 ) ∧ 𝐹 ≠ 𝐻 ) ) → Fun ( ◡ { ⟨ 𝐴 , 𝐵 ⟩ , ⟨ 𝐶 , 𝐷 ⟩ } ∪ ◡ { ⟨ 𝐸 , 𝐹 ⟩ , ⟨ 𝐺 , 𝐻 ⟩ } ) )
24		cnvun	⊢ ◡ ( { ⟨ 𝐴 , 𝐵 ⟩ , ⟨ 𝐶 , 𝐷 ⟩ } ∪ { ⟨ 𝐸 , 𝐹 ⟩ , ⟨ 𝐺 , 𝐻 ⟩ } ) = ( ◡ { ⟨ 𝐴 , 𝐵 ⟩ , ⟨ 𝐶 , 𝐷 ⟩ } ∪ ◡ { ⟨ 𝐸 , 𝐹 ⟩ , ⟨ 𝐺 , 𝐻 ⟩ } )
25	24	funeqi	⊢ ( Fun ◡ ( { ⟨ 𝐴 , 𝐵 ⟩ , ⟨ 𝐶 , 𝐷 ⟩ } ∪ { ⟨ 𝐸 , 𝐹 ⟩ , ⟨ 𝐺 , 𝐻 ⟩ } ) ↔ Fun ( ◡ { ⟨ 𝐴 , 𝐵 ⟩ , ⟨ 𝐶 , 𝐷 ⟩ } ∪ ◡ { ⟨ 𝐸 , 𝐹 ⟩ , ⟨ 𝐺 , 𝐻 ⟩ } ) )
26	23 25	sylibr	⊢ ( ( ( ( 𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ) ∧ ( 𝐸 ∈ 𝑊 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( ( 𝐵 ≠ 𝐷 ∧ 𝐵 ≠ 𝐹 ∧ 𝐵 ≠ 𝐻 ) ∧ ( 𝐷 ≠ 𝐹 ∧ 𝐷 ≠ 𝐻 ) ∧ 𝐹 ≠ 𝐻 ) ) → Fun ◡ ( { ⟨ 𝐴 , 𝐵 ⟩ , ⟨ 𝐶 , 𝐷 ⟩ } ∪ { ⟨ 𝐸 , 𝐹 ⟩ , ⟨ 𝐺 , 𝐻 ⟩ } ) )