hdmap1cbv - Mathbox for Norm Megill

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Theorem hdmap1cbv 37092

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 6191	. . . . 5 ⊢ (𝑥 = 𝑦 → (2^nd ‘𝑥) = (2^nd ‘𝑦))
3	2	eqeq1d 2624	. . . 4 ⊢ (𝑥 = 𝑦 → ((2^nd ‘𝑥) = 0 ↔ (2^nd ‘𝑦) = 0 ))
4	2	sneqd 4189	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → {(2^nd ‘𝑥)} = {(2^nd ‘𝑦)})
5	4	fveq2d 6195	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑁‘{(2^nd ‘𝑥)}) = (𝑁‘{(2^nd ‘𝑦)}))
6	5	fveq2d 6195	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝑀‘(𝑁‘{(2^nd ‘𝑦)})))
7	6	eqeq1d 2624	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ↔ (𝑀‘(𝑁‘{(2^nd ‘𝑦)})) = (𝐽‘{ℎ})))
8		fveq2 6191	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (1^st ‘𝑥) = (1^st ‘𝑦))
9	8	fveq2d 6195	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (1^st ‘(1^st ‘𝑥)) = (1^st ‘(1^st ‘𝑦)))
10	9, 2	oveq12d 6668	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥)) = ((1^st ‘(1^st ‘𝑦)) − (2^nd ‘𝑦)))
11	10	sneqd 4189	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → {((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))} = {((1^st ‘(1^st ‘𝑦)) − (2^nd ‘𝑦))})
12	11	fveq2d 6195	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))}) = (𝑁‘{((1^st ‘(1^st ‘𝑦)) − (2^nd ‘𝑦))}))
13	12	fveq2d 6195	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑦)) − (2^nd ‘𝑦))})))
14	8	fveq2d 6195	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (2^nd ‘(1^st ‘𝑥)) = (2^nd ‘(1^st ‘𝑦)))
15	14	oveq1d 6665	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((2^nd ‘(1^st ‘𝑥))𝑅ℎ) = ((2^nd ‘(1^st ‘𝑦))𝑅ℎ))
16	15	sneqd 4189	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → {((2^nd ‘(1^st ‘𝑥))𝑅ℎ)} = {((2^nd ‘(1^st ‘𝑦))𝑅ℎ)})
17	16	fveq2d 6195	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)}) = (𝐽‘{((2^nd ‘(1^st ‘𝑦))𝑅ℎ)}))
18	13, 17	eqeq12d 2637	. . . . . 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)})))
19	7, 18	anbi12d 747	. . . . 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)}))))
20	19	riotabidv 6613	. . . 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)}))))
21	3, 20	ifbieq2d 4111	. . 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)})))))
22	21	cbvmptv 4750	. 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)})))))
23		sneq 4187	. . . . . . . 8 ⊢ (ℎ = 𝑖 → {ℎ} = {𝑖})
24	23	fveq2d 6195	. . . . . . 7 ⊢ (ℎ = 𝑖 → (𝐽‘{ℎ}) = (𝐽‘{𝑖}))
25	24	eqeq2d 2632	. . . . . 6 ⊢ (ℎ = 𝑖 → ((𝑀‘(𝑁‘{(2^nd ‘𝑦)})) = (𝐽‘{ℎ}) ↔ (𝑀‘(𝑁‘{(2^nd ‘𝑦)})) = (𝐽‘{𝑖})))
26		oveq2 6658	. . . . . . . . 9 ⊢ (ℎ = 𝑖 → ((2^nd ‘(1^st ‘𝑦))𝑅ℎ) = ((2^nd ‘(1^st ‘𝑦))𝑅𝑖))
27	26	sneqd 4189	. . . . . . . 8 ⊢ (ℎ = 𝑖 → {((2^nd ‘(1^st ‘𝑦))𝑅ℎ)} = {((2^nd ‘(1^st ‘𝑦))𝑅𝑖)})
28	27	fveq2d 6195	. . . . . . 7 ⊢ (ℎ = 𝑖 → (𝐽‘{((2^nd ‘(1^st ‘𝑦))𝑅ℎ)}) = (𝐽‘{((2^nd ‘(1^st ‘𝑦))𝑅𝑖)}))
29	28	eqeq2d 2632	. . . . . 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)})))
30	25, 29	anbi12d 747	. . . . 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)}))))
31	30	cbvriotav 6622	. . . 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)})))
32		ifeq2 4091	. . . 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)})))))
33	31, 32	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)}))))
34	33	mpteq2i 4741	. 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)})))))
35	1, 22, 34	3eqtri 2648	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 384 = wceq 1483 Vcvv 3200 ifcif 4086 {csn 4177 ↦ cmpt 4729 ‘cfv 5888 ℩crio 6610 (class class class)co 6650 1^st c1st 7166 2^nd c2nd 7167

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-iota 5851 df-fv 5896 df-riota 6611 df-ov 6653

This theorem is referenced by: hdmap1valc 37093 hdmap1eu 37115 hdmap1euOLDN 37116

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