hdmap1cbv - Mathbox for Norm Megill

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Theorem hdmap1cbv 42438

Description: Frequently used lemma to change bound variables in 𝐿 hypothesis. (Contributed by NM, 15-May-2015.)

**Hypothesis**
Ref	Expression
hdmap1cbv.l	⊢ 𝐿 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)})))))

**Assertion**
Ref	Expression
hdmap1cbv	⊢ 𝐿 = (𝑦 ∈ V ↦ if((2^nd ‘𝑦) = 0 , 𝑄, (℩𝑖 ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑦)})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑦)) − (2^nd ‘𝑦))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑦))𝑅𝑖)})))))

Distinct variable groups: ℎ,𝑖,𝑥,𝑦,𝐷 ℎ,𝐽,𝑖,𝑥,𝑦 ℎ,𝑀,𝑖,𝑥,𝑦 ℎ,𝑁,𝑖,𝑥,𝑦 𝑥, 0 ,𝑦 𝑥,𝑄,𝑦 𝑅,ℎ,𝑖,𝑥,𝑦 − ,ℎ,𝑖,𝑥,𝑦 Allowed substitution hints: 𝑄(ℎ,𝑖) 𝐿(𝑥,𝑦,ℎ,𝑖) 0 (ℎ,𝑖)

**Proof of Theorem hdmap1cbv**
Step	Hyp	Ref	Expression
1		hdmap1cbv.l	. 2 ⊢ 𝐿 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)})))))
2		fveq2 6871	. . . . 5 ⊢ (𝑥 = 𝑦 → (2^nd ‘𝑥) = (2^nd ‘𝑦))
3	2	eqeq1d 2767	. . . 4 ⊢ (𝑥 = 𝑦 → ((2^nd ‘𝑥) = 0 ↔ (2^nd ‘𝑦) = 0 ))
4	2	sneqd 4597	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → {(2^nd ‘𝑥)} = {(2^nd ‘𝑦)})
5	4	fveq2d 6875	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑁‘{(2^nd ‘𝑥)}) = (𝑁‘{(2^nd ‘𝑦)}))
6	5	fveq2d 6875	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝑀‘(𝑁‘{(2^nd ‘𝑦)})))
7	6	eqeq1d 2767	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ↔ (𝑀‘(𝑁‘{(2^nd ‘𝑦)})) = (𝐽‘{ℎ})))
8		2fveq3 6876	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (1^st ‘(1^st ‘𝑥)) = (1^st ‘(1^st ‘𝑦)))
9	8, 2	oveq12d 7418	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥)) = ((1^st ‘(1^st ‘𝑦)) − (2^nd ‘𝑦)))
10	9	sneqd 4597	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → {((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))} = {((1^st ‘(1^st ‘𝑦)) − (2^nd ‘𝑦))})
11	10	fveq2d 6875	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))}) = (𝑁‘{((1^st ‘(1^st ‘𝑦)) − (2^nd ‘𝑦))}))
12	11	fveq2d 6875	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑦)) − (2^nd ‘𝑦))})))
13		2fveq3 6876	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (2^nd ‘(1^st ‘𝑥)) = (2^nd ‘(1^st ‘𝑦)))
14	13	oveq1d 7415	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((2^nd ‘(1^st ‘𝑥))𝑅ℎ) = ((2^nd ‘(1^st ‘𝑦))𝑅ℎ))
15	14	sneqd 4597	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → {((2^nd ‘(1^st ‘𝑥))𝑅ℎ)} = {((2^nd ‘(1^st ‘𝑦))𝑅ℎ)})
16	15	fveq2d 6875	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)}) = (𝐽‘{((2^nd ‘(1^st ‘𝑦))𝑅ℎ)}))
17	12, 16	eqeq12d 2781	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)}) ↔ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑦)) − (2^nd ‘𝑦))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑦))𝑅ℎ)})))
18	7, 17	anbi12d 643	. . . . 5 ⊢ (𝑥 = 𝑦 → (((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)})) ↔ ((𝑀‘(𝑁‘{(2^nd ‘𝑦)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑦)) − (2^nd ‘𝑦))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑦))𝑅ℎ)}))))
19	18	riotabidv 7359	. . . 4 ⊢ (𝑥 = 𝑦 → (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)}))) = (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑦)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑦)) − (2^nd ‘𝑦))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑦))𝑅ℎ)}))))
20	3, 19	ifbieq2d 4510	. . 3 ⊢ (𝑥 = 𝑦 → if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)})))) = if((2^nd ‘𝑦) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑦)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑦)) − (2^nd ‘𝑦))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑦))𝑅ℎ)})))))
21	20	cbvmptv 5209	. 2 ⊢ (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)}))))) = (𝑦 ∈ V ↦ if((2^nd ‘𝑦) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑦)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑦)) − (2^nd ‘𝑦))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑦))𝑅ℎ)})))))
22		sneq 4595	. . . . . . . 8 ⊢ (ℎ = 𝑖 → {ℎ} = {𝑖})
23	22	fveq2d 6875	. . . . . . 7 ⊢ (ℎ = 𝑖 → (𝐽‘{ℎ}) = (𝐽‘{𝑖}))
24	23	eqeq2d 2776	. . . . . 6 ⊢ (ℎ = 𝑖 → ((𝑀‘(𝑁‘{(2^nd ‘𝑦)})) = (𝐽‘{ℎ}) ↔ (𝑀‘(𝑁‘{(2^nd ‘𝑦)})) = (𝐽‘{𝑖})))
25		oveq2 7408	. . . . . . . . 9 ⊢ (ℎ = 𝑖 → ((2^nd ‘(1^st ‘𝑦))𝑅ℎ) = ((2^nd ‘(1^st ‘𝑦))𝑅𝑖))
26	25	sneqd 4597	. . . . . . . 8 ⊢ (ℎ = 𝑖 → {((2^nd ‘(1^st ‘𝑦))𝑅ℎ)} = {((2^nd ‘(1^st ‘𝑦))𝑅𝑖)})
27	26	fveq2d 6875	. . . . . . 7 ⊢ (ℎ = 𝑖 → (𝐽‘{((2^nd ‘(1^st ‘𝑦))𝑅ℎ)}) = (𝐽‘{((2^nd ‘(1^st ‘𝑦))𝑅𝑖)}))
28	27	eqeq2d 2776	. . . . . 6 ⊢ (ℎ = 𝑖 → ((𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑦)) − (2^nd ‘𝑦))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑦))𝑅ℎ)}) ↔ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑦)) − (2^nd ‘𝑦))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑦))𝑅𝑖)})))
29	24, 28	anbi12d 643	. . . . 5 ⊢ (ℎ = 𝑖 → (((𝑀‘(𝑁‘{(2^nd ‘𝑦)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑦)) − (2^nd ‘𝑦))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑦))𝑅ℎ)})) ↔ ((𝑀‘(𝑁‘{(2^nd ‘𝑦)})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑦)) − (2^nd ‘𝑦))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑦))𝑅𝑖)}))))
30	29	cbvriotavw 7367	. . . 4 ⊢ (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑦)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑦)) − (2^nd ‘𝑦))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑦))𝑅ℎ)}))) = (℩𝑖 ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑦)})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑦)) − (2^nd ‘𝑦))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑦))𝑅𝑖)})))
31		ifeq2 4488	. . . 4 ⊢ ((℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑦)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑦)) − (2^nd ‘𝑦))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑦))𝑅ℎ)}))) = (℩𝑖 ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑦)})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑦)) − (2^nd ‘𝑦))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑦))𝑅𝑖)}))) → if((2^nd ‘𝑦) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑦)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑦)) − (2^nd ‘𝑦))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑦))𝑅ℎ)})))) = if((2^nd ‘𝑦) = 0 , 𝑄, (℩𝑖 ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑦)})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑦)) − (2^nd ‘𝑦))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑦))𝑅𝑖)})))))
32	30, 31	ax-mp 5	. . 3 ⊢ if((2^nd ‘𝑦) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑦)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑦)) − (2^nd ‘𝑦))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑦))𝑅ℎ)})))) = if((2^nd ‘𝑦) = 0 , 𝑄, (℩𝑖 ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑦)})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑦)) − (2^nd ‘𝑦))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑦))𝑅𝑖)}))))
33	32	mpteq2i 5201	. 2 ⊢ (𝑦 ∈ V ↦ if((2^nd ‘𝑦) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑦)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑦)) − (2^nd ‘𝑦))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑦))𝑅ℎ)}))))) = (𝑦 ∈ V ↦ if((2^nd ‘𝑦) = 0 , 𝑄, (℩𝑖 ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑦)})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑦)) − (2^nd ‘𝑦))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑦))𝑅𝑖)})))))
34	1, 21, 33	3eqtri 2792	1 ⊢ 𝐿 = (𝑦 ∈ V ↦ if((2^nd ‘𝑦) = 0 , 𝑄, (℩𝑖 ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑦)})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑦)) − (2^nd ‘𝑦))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑦))𝑅𝑖)})))))

Colors of variables: wff setvar class

Syntax hints: ∧ wa 400 = wceq 1563 Vcvv 3457 ifcif 4483 {csn 4585 ↦ cmpt 5186 ‘cfv 6525 ℩crio 7356 (class class class)co 7400 1^st c1st 7972 2^nd c2nd 7973

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-ext 2737

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-sb 2094 df-clab 2744 df-cleq 2757 df-clel 2840 df-rab 3418 df-v 3459 df-dif 3910 df-un 3912 df-ss 3924 df-nul 4289 df-if 4484 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4869 df-br 5106 df-opab 5168 df-mpt 5187 df-iota 6481 df-fv 6533 df-riota 7357 df-ov 7403

This theorem is referenced by: hdmap1valc 42439 hdmap1eu 42460 hdmap1euOLDN 42461

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