Theorem mpoeq123 5901

Description: An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) (Revised by Mario Carneiro, 19-Mar-2015.)

Assertion

Ref	Expression
mpoeq123	⊢ ((𝐴 = 𝐷 ∧ ∀𝑥 ∈ 𝐴 (𝐵 = 𝐸 ∧ ∀𝑦 ∈ 𝐵 𝐶 = 𝐹)) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐷, 𝑦 ∈ 𝐸 ↦ 𝐹))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝑥,𝐷,𝑦   𝑦,𝐸

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥,𝑦)   𝐸(𝑥)   𝐹(𝑥,𝑦)

Proof of Theorem mpoeq123

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1516	. . . 4 ⊢ Ⅎ𝑥 𝐴 = 𝐷
2		nfra1 2497	. . . 4 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 (𝐵 = 𝐸 ∧ ∀𝑦 ∈ 𝐵 𝐶 = 𝐹)
3	1, 2	nfan 1553	. . 3 ⊢ Ⅎ𝑥(𝐴 = 𝐷 ∧ ∀𝑥 ∈ 𝐴 (𝐵 = 𝐸 ∧ ∀𝑦 ∈ 𝐵 𝐶 = 𝐹))
4		nfv 1516	. . . 4 ⊢ Ⅎ𝑦 𝐴 = 𝐷
5		nfcv 2308	. . . . 5 ⊢ Ⅎ𝑦𝐴
6		nfv 1516	. . . . . 6 ⊢ Ⅎ𝑦 𝐵 = 𝐸
7		nfra1 2497	. . . . . 6 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐵 𝐶 = 𝐹
8	6, 7	nfan 1553	. . . . 5 ⊢ Ⅎ𝑦(𝐵 = 𝐸 ∧ ∀𝑦 ∈ 𝐵 𝐶 = 𝐹)
9	5, 8	nfralxy 2504	. . . 4 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 (𝐵 = 𝐸 ∧ ∀𝑦 ∈ 𝐵 𝐶 = 𝐹)
10	4, 9	nfan 1553	. . 3 ⊢ Ⅎ𝑦(𝐴 = 𝐷 ∧ ∀𝑥 ∈ 𝐴 (𝐵 = 𝐸 ∧ ∀𝑦 ∈ 𝐵 𝐶 = 𝐹))
11		nfv 1516	. . 3 ⊢ Ⅎ𝑧(𝐴 = 𝐷 ∧ ∀𝑥 ∈ 𝐴 (𝐵 = 𝐸 ∧ ∀𝑦 ∈ 𝐵 𝐶 = 𝐹))
12		rsp 2513	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 (𝐵 = 𝐸 ∧ ∀𝑦 ∈ 𝐵 𝐶 = 𝐹) → (𝑥 ∈ 𝐴 → (𝐵 = 𝐸 ∧ ∀𝑦 ∈ 𝐵 𝐶 = 𝐹)))
13		rsp 2513	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝐵 𝐶 = 𝐹 → (𝑦 ∈ 𝐵 → 𝐶 = 𝐹))
14		eqeq2 2175	. . . . . . . . . 10 ⊢ (𝐶 = 𝐹 → (𝑧 = 𝐶 ↔ 𝑧 = 𝐹))
15	13, 14	syl6 33	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐵 𝐶 = 𝐹 → (𝑦 ∈ 𝐵 → (𝑧 = 𝐶 ↔ 𝑧 = 𝐹)))
16	15	pm5.32d 446	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝐵 𝐶 = 𝐹 → ((𝑦 ∈ 𝐵 ∧ 𝑧 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑧 = 𝐹)))
17		eleq2 2230	. . . . . . . . 9 ⊢ (𝐵 = 𝐸 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐸))
18	17	anbi1d 461	. . . . . . . 8 ⊢ (𝐵 = 𝐸 → ((𝑦 ∈ 𝐵 ∧ 𝑧 = 𝐹) ↔ (𝑦 ∈ 𝐸 ∧ 𝑧 = 𝐹)))
19	16, 18	sylan9bbr 459	. . . . . . 7 ⊢ ((𝐵 = 𝐸 ∧ ∀𝑦 ∈ 𝐵 𝐶 = 𝐹) → ((𝑦 ∈ 𝐵 ∧ 𝑧 = 𝐶) ↔ (𝑦 ∈ 𝐸 ∧ 𝑧 = 𝐹)))
20	12, 19	syl6 33	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (𝐵 = 𝐸 ∧ ∀𝑦 ∈ 𝐵 𝐶 = 𝐹) → (𝑥 ∈ 𝐴 → ((𝑦 ∈ 𝐵 ∧ 𝑧 = 𝐶) ↔ (𝑦 ∈ 𝐸 ∧ 𝑧 = 𝐹))))
21	20	pm5.32d 446	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝐵 = 𝐸 ∧ ∀𝑦 ∈ 𝐵 𝐶 = 𝐹) → ((𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 = 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐸 ∧ 𝑧 = 𝐹))))
22		eleq2 2230	. . . . . 6 ⊢ (𝐴 = 𝐷 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐷))
23	22	anbi1d 461	. . . . 5 ⊢ (𝐴 = 𝐷 → ((𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐸 ∧ 𝑧 = 𝐹)) ↔ (𝑥 ∈ 𝐷 ∧ (𝑦 ∈ 𝐸 ∧ 𝑧 = 𝐹))))
24	21, 23	sylan9bbr 459	. . . 4 ⊢ ((𝐴 = 𝐷 ∧ ∀𝑥 ∈ 𝐴 (𝐵 = 𝐸 ∧ ∀𝑦 ∈ 𝐵 𝐶 = 𝐹)) → ((𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 = 𝐶)) ↔ (𝑥 ∈ 𝐷 ∧ (𝑦 ∈ 𝐸 ∧ 𝑧 = 𝐹))))
25		anass 399	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 = 𝐶)))
26		anass 399	. . . 4 ⊢ (((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸) ∧ 𝑧 = 𝐹) ↔ (𝑥 ∈ 𝐷 ∧ (𝑦 ∈ 𝐸 ∧ 𝑧 = 𝐹)))
27	24, 25, 26	3bitr4g 222	. . 3 ⊢ ((𝐴 = 𝐷 ∧ ∀𝑥 ∈ 𝐴 (𝐵 = 𝐸 ∧ ∀𝑦 ∈ 𝐵 𝐶 = 𝐹)) → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸) ∧ 𝑧 = 𝐹)))
28	3, 10, 11, 27	oprabbid 5895	. 2 ⊢ ((𝐴 = 𝐷 ∧ ∀𝑥 ∈ 𝐴 (𝐵 = 𝐸 ∧ ∀𝑦 ∈ 𝐵 𝐶 = 𝐹)) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸) ∧ 𝑧 = 𝐹)})
29		df-mpo 5847	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
30		df-mpo 5847	. 2 ⊢ (𝑥 ∈ 𝐷, 𝑦 ∈ 𝐸 ↦ 𝐹) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸) ∧ 𝑧 = 𝐹)}
31	28, 29, 30	3eqtr4g 2224	1 ⊢ ((𝐴 = 𝐷 ∧ ∀𝑥 ∈ 𝐴 (𝐵 = 𝐸 ∧ ∀𝑦 ∈ 𝐵 𝐶 = 𝐹)) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐷, 𝑦 ∈ 𝐸 ↦ 𝐹))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1343   ∈ wcel 2136  ∀wral 2444  {coprab 5843   ∈ cmpo 5844

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-oprab 5846  df-mpo 5847

This theorem is referenced by:  mpoeq12  5902  mapxpen  6814