ovg

Description: The value of an operation class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010)

	Ref	Expression
Hypotheses	ovg.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
	ovg.2	⊢ ( 𝑦 = 𝐵 → ( 𝜓 ↔ 𝜒 ) )
	ovg.3	⊢ ( 𝑧 = 𝐶 → ( 𝜒 ↔ 𝜃 ) )
	ovg.4	⊢ ( ( 𝜏 ∧ ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ) → ∃! 𝑧 𝜑 )
	ovg.5	⊢ 𝐹 = { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) }
Assertion	ovg	⊢ ( ( 𝜏 ∧ ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝐷 ) ) → ( ( 𝐴 𝐹 𝐵 ) = 𝐶 ↔ 𝜃 ) )

Proof

Step	Hyp	Ref	Expression
1		ovg.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
2		ovg.2	⊢ ( 𝑦 = 𝐵 → ( 𝜓 ↔ 𝜒 ) )
3		ovg.3	⊢ ( 𝑧 = 𝐶 → ( 𝜒 ↔ 𝜃 ) )
4		ovg.4	⊢ ( ( 𝜏 ∧ ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ) → ∃! 𝑧 𝜑 )
5		ovg.5	⊢ 𝐹 = { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) }
6		df-ov	⊢ ( 𝐴 𝐹 𝐵 ) = ( 𝐹 ‘ ⟨ 𝐴 , 𝐵 ⟩ )
7	5	fveq1i	⊢ ( 𝐹 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = ( { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) } ‘ ⟨ 𝐴 , 𝐵 ⟩ )
8	6 7	eqtri	⊢ ( 𝐴 𝐹 𝐵 ) = ( { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) } ‘ ⟨ 𝐴 , 𝐵 ⟩ )
9	8	eqeq1i	⊢ ( ( 𝐴 𝐹 𝐵 ) = 𝐶 ↔ ( { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) } ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = 𝐶 )
10		eqeq2	⊢ ( 𝑐 = 𝐶 → ( ( { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) } ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = 𝑐 ↔ ( { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) } ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = 𝐶 ) )
11		opeq2	⊢ ( 𝑐 = 𝐶 → ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝑐 ⟩ = ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶 ⟩ )
12	11	eleq1d	⊢ ( 𝑐 = 𝐶 → ( ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝑐 ⟩ ∈ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) } ↔ ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶 ⟩ ∈ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) } ) )
13	10 12	bibi12d	⊢ ( 𝑐 = 𝐶 → ( ( ( { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) } ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = 𝑐 ↔ ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝑐 ⟩ ∈ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) } ) ↔ ( ( { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) } ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = 𝐶 ↔ ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶 ⟩ ∈ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) } ) ) )
14	13	imbi2d	⊢ ( 𝑐 = 𝐶 → ( ( ( 𝜏 ∧ ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ) → ( ( { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) } ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = 𝑐 ↔ ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝑐 ⟩ ∈ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) } ) ) ↔ ( ( 𝜏 ∧ ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ) → ( ( { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) } ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = 𝐶 ↔ ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶 ⟩ ∈ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) } ) ) ) )
15	4	ex	⊢ ( 𝜏 → ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) → ∃! 𝑧 𝜑 ) )
16	15	alrimivv	⊢ ( 𝜏 → ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) → ∃! 𝑧 𝜑 ) )
17		fnoprabg	⊢ ( ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) → ∃! 𝑧 𝜑 ) → { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) } Fn { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) } )
18	16 17	syl	⊢ ( 𝜏 → { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) } Fn { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) } )
19		eleq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 ∈ 𝑅 ↔ 𝐴 ∈ 𝑅 ) )
20	19	anbi1d	⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ↔ ( 𝐴 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ) )
21		eleq1	⊢ ( 𝑦 = 𝐵 → ( 𝑦 ∈ 𝑆 ↔ 𝐵 ∈ 𝑆 ) )
22	21	anbi2d	⊢ ( 𝑦 = 𝐵 → ( ( 𝐴 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ↔ ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ) )
23	20 22	opelopabg	⊢ ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) → ( ⟨ 𝐴 , 𝐵 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) } ↔ ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ) )
24	23	ibir	⊢ ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) → ⟨ 𝐴 , 𝐵 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) } )
25		fnopfvb	⊢ ( ( { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) } Fn { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) } ∧ ⟨ 𝐴 , 𝐵 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) } ) → ( ( { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) } ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = 𝑐 ↔ ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝑐 ⟩ ∈ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) } ) )
26	18 24 25	syl2an	⊢ ( ( 𝜏 ∧ ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ) → ( ( { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) } ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = 𝑐 ↔ ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝑐 ⟩ ∈ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) } ) )
27	14 26	vtoclg	⊢ ( 𝐶 ∈ 𝐷 → ( ( 𝜏 ∧ ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ) → ( ( { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) } ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = 𝐶 ↔ ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶 ⟩ ∈ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) } ) ) )
28	27	com12	⊢ ( ( 𝜏 ∧ ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ) → ( 𝐶 ∈ 𝐷 → ( ( { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) } ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = 𝐶 ↔ ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶 ⟩ ∈ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) } ) ) )
29	28	exp32	⊢ ( 𝜏 → ( 𝐴 ∈ 𝑅 → ( 𝐵 ∈ 𝑆 → ( 𝐶 ∈ 𝐷 → ( ( { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) } ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = 𝐶 ↔ ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶 ⟩ ∈ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) } ) ) ) ) )
30	29	3imp2	⊢ ( ( 𝜏 ∧ ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝐷 ) ) → ( ( { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) } ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = 𝐶 ↔ ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶 ⟩ ∈ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) } ) )
31	20 1	anbi12d	⊢ ( 𝑥 = 𝐴 → ( ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) ↔ ( ( 𝐴 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜓 ) ) )
32	22 2	anbi12d	⊢ ( 𝑦 = 𝐵 → ( ( ( 𝐴 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜓 ) ↔ ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ 𝜒 ) ) )
33	3	anbi2d	⊢ ( 𝑧 = 𝐶 → ( ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ 𝜒 ) ↔ ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ 𝜃 ) ) )
34	31 32 33	eloprabg	⊢ ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝐷 ) → ( ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶 ⟩ ∈ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) } ↔ ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ 𝜃 ) ) )
35	34	adantl	⊢ ( ( 𝜏 ∧ ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝐷 ) ) → ( ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶 ⟩ ∈ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) } ↔ ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ 𝜃 ) ) )
36	30 35	bitrd	⊢ ( ( 𝜏 ∧ ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝐷 ) ) → ( ( { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) } ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = 𝐶 ↔ ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ 𝜃 ) ) )
37	9 36	bitrid	⊢ ( ( 𝜏 ∧ ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝐷 ) ) → ( ( 𝐴 𝐹 𝐵 ) = 𝐶 ↔ ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ 𝜃 ) ) )
38		biidd	⊢ ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) → ( ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ 𝜃 ) ↔ ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ 𝜃 ) ) )
39	38	bianabs	⊢ ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) → ( ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ 𝜃 ) ↔ 𝜃 ) )
40	39	3adant3	⊢ ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝐷 ) → ( ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ 𝜃 ) ↔ 𝜃 ) )
41	40	adantl	⊢ ( ( 𝜏 ∧ ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝐷 ) ) → ( ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ 𝜃 ) ↔ 𝜃 ) )
42	37 41	bitrd	⊢ ( ( 𝜏 ∧ ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝐷 ) ) → ( ( 𝐴 𝐹 𝐵 ) = 𝐶 ↔ 𝜃 ) )

Metamath Proof Explorer

Theorem ovg

Proof