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Theorem hdmap1cbv 39461

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 6676	. . . . 5 ⊢ (𝑥 = 𝑦 → (2^nd ‘𝑥) = (2^nd ‘𝑦))
3	2	eqeq1d 2740	. . . 4 ⊢ (𝑥 = 𝑦 → ((2^nd ‘𝑥) = 0 ↔ (2^nd ‘𝑦) = 0 ))
4	2	sneqd 4528	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → {(2^nd ‘𝑥)} = {(2^nd ‘𝑦)})
5	4	fveq2d 6680	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑁‘{(2^nd ‘𝑥)}) = (𝑁‘{(2^nd ‘𝑦)}))
6	5	fveq2d 6680	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝑀‘(𝑁‘{(2^nd ‘𝑦)})))
7	6	eqeq1d 2740	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ↔ (𝑀‘(𝑁‘{(2^nd ‘𝑦)})) = (𝐽‘{ℎ})))
8		2fveq3 6681	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (1^st ‘(1^st ‘𝑥)) = (1^st ‘(1^st ‘𝑦)))
9	8, 2	oveq12d 7190	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥)) = ((1^st ‘(1^st ‘𝑦)) − (2^nd ‘𝑦)))
10	9	sneqd 4528	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → {((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))} = {((1^st ‘(1^st ‘𝑦)) − (2^nd ‘𝑦))})
11	10	fveq2d 6680	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))}) = (𝑁‘{((1^st ‘(1^st ‘𝑦)) − (2^nd ‘𝑦))}))
12	11	fveq2d 6680	. . . . . . 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 6681	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (2^nd ‘(1^st ‘𝑥)) = (2^nd ‘(1^st ‘𝑦)))
14	13	oveq1d 7187	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((2^nd ‘(1^st ‘𝑥))𝑅ℎ) = ((2^nd ‘(1^st ‘𝑦))𝑅ℎ))
15	14	sneqd 4528	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → {((2^nd ‘(1^st ‘𝑥))𝑅ℎ)} = {((2^nd ‘(1^st ‘𝑦))𝑅ℎ)})
16	15	fveq2d 6680	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)}) = (𝐽‘{((2^nd ‘(1^st ‘𝑦))𝑅ℎ)}))
17	12, 16	eqeq12d 2754	. . . . . 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 634	. . . . 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 7131	. . . 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 4440	. . 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 5133	. 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 4526	. . . . . . . 8 ⊢ (ℎ = 𝑖 → {ℎ} = {𝑖})
23	22	fveq2d 6680	. . . . . . 7 ⊢ (ℎ = 𝑖 → (𝐽‘{ℎ}) = (𝐽‘{𝑖}))
24	23	eqeq2d 2749	. . . . . 6 ⊢ (ℎ = 𝑖 → ((𝑀‘(𝑁‘{(2^nd ‘𝑦)})) = (𝐽‘{ℎ}) ↔ (𝑀‘(𝑁‘{(2^nd ‘𝑦)})) = (𝐽‘{𝑖})))
25		oveq2 7180	. . . . . . . . 9 ⊢ (ℎ = 𝑖 → ((2^nd ‘(1^st ‘𝑦))𝑅ℎ) = ((2^nd ‘(1^st ‘𝑦))𝑅𝑖))
26	25	sneqd 4528	. . . . . . . 8 ⊢ (ℎ = 𝑖 → {((2^nd ‘(1^st ‘𝑦))𝑅ℎ)} = {((2^nd ‘(1^st ‘𝑦))𝑅𝑖)})
27	26	fveq2d 6680	. . . . . . 7 ⊢ (ℎ = 𝑖 → (𝐽‘{((2^nd ‘(1^st ‘𝑦))𝑅ℎ)}) = (𝐽‘{((2^nd ‘(1^st ‘𝑦))𝑅𝑖)}))
28	27	eqeq2d 2749	. . . . . 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 634	. . . . 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 7139	. . . 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 4419	. . . 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 5122	. 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 2765	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 399 = wceq 1542 Vcvv 3398 ifcif 4414 {csn 4516 ↦ cmpt 5110 ‘cfv 6339 ℩crio 7128 (class class class)co 7172 1^st c1st 7714 2^nd c2nd 7715

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-ex 1787 df-nf 1791 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ral 3058 df-rab 3062 df-v 3400 df-un 3848 df-in 3850 df-ss 3860 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-mpt 5111 df-iota 6297 df-fv 6347 df-riota 7129 df-ov 7175

This theorem is referenced by: hdmap1valc 39462 hdmap1eu 39483 hdmap1euOLDN 39484

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