Theorem mpoeq123 7226

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 1915	. . . 4 ⊢ Ⅎ𝑥 𝐴 = 𝐷
2		nfra1 3219	. . . 4 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 (𝐵 = 𝐸 ∧ ∀𝑦 ∈ 𝐵 𝐶 = 𝐹)
3	1, 2	nfan 1900	. . 3 ⊢ Ⅎ𝑥(𝐴 = 𝐷 ∧ ∀𝑥 ∈ 𝐴 (𝐵 = 𝐸 ∧ ∀𝑦 ∈ 𝐵 𝐶 = 𝐹))
4		nfv 1915	. . . 4 ⊢ Ⅎ𝑦 𝐴 = 𝐷
5		nfcv 2977	. . . . 5 ⊢ Ⅎ𝑦𝐴
6		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑦 𝐵 = 𝐸
7		nfra1 3219	. . . . . 6 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐵 𝐶 = 𝐹
8	6, 7	nfan 1900	. . . . 5 ⊢ Ⅎ𝑦(𝐵 = 𝐸 ∧ ∀𝑦 ∈ 𝐵 𝐶 = 𝐹)
9	5, 8	nfralw 3225	. . . 4 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 (𝐵 = 𝐸 ∧ ∀𝑦 ∈ 𝐵 𝐶 = 𝐹)
10	4, 9	nfan 1900	. . 3 ⊢ Ⅎ𝑦(𝐴 = 𝐷 ∧ ∀𝑥 ∈ 𝐴 (𝐵 = 𝐸 ∧ ∀𝑦 ∈ 𝐵 𝐶 = 𝐹))
11		nfv 1915	. . 3 ⊢ Ⅎ𝑧(𝐴 = 𝐷 ∧ ∀𝑥 ∈ 𝐴 (𝐵 = 𝐸 ∧ ∀𝑦 ∈ 𝐵 𝐶 = 𝐹))
12		rsp 3205	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 (𝐵 = 𝐸 ∧ ∀𝑦 ∈ 𝐵 𝐶 = 𝐹) → (𝑥 ∈ 𝐴 → (𝐵 = 𝐸 ∧ ∀𝑦 ∈ 𝐵 𝐶 = 𝐹)))
13		rsp 3205	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝐵 𝐶 = 𝐹 → (𝑦 ∈ 𝐵 → 𝐶 = 𝐹))
14		eqeq2 2833	. . . . . . . . . 10 ⊢ (𝐶 = 𝐹 → (𝑧 = 𝐶 ↔ 𝑧 = 𝐹))
15	13, 14	syl6 35	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐵 𝐶 = 𝐹 → (𝑦 ∈ 𝐵 → (𝑧 = 𝐶 ↔ 𝑧 = 𝐹)))
16	15	pm5.32d 579	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝐵 𝐶 = 𝐹 → ((𝑦 ∈ 𝐵 ∧ 𝑧 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑧 = 𝐹)))
17		eleq2 2901	. . . . . . . . 9 ⊢ (𝐵 = 𝐸 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐸))
18	17	anbi1d 631	. . . . . . . 8 ⊢ (𝐵 = 𝐸 → ((𝑦 ∈ 𝐵 ∧ 𝑧 = 𝐹) ↔ (𝑦 ∈ 𝐸 ∧ 𝑧 = 𝐹)))
19	16, 18	sylan9bbr 513	. . . . . . 7 ⊢ ((𝐵 = 𝐸 ∧ ∀𝑦 ∈ 𝐵 𝐶 = 𝐹) → ((𝑦 ∈ 𝐵 ∧ 𝑧 = 𝐶) ↔ (𝑦 ∈ 𝐸 ∧ 𝑧 = 𝐹)))
20	12, 19	syl6 35	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (𝐵 = 𝐸 ∧ ∀𝑦 ∈ 𝐵 𝐶 = 𝐹) → (𝑥 ∈ 𝐴 → ((𝑦 ∈ 𝐵 ∧ 𝑧 = 𝐶) ↔ (𝑦 ∈ 𝐸 ∧ 𝑧 = 𝐹))))
21	20	pm5.32d 579	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝐵 = 𝐸 ∧ ∀𝑦 ∈ 𝐵 𝐶 = 𝐹) → ((𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 = 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐸 ∧ 𝑧 = 𝐹))))
22		eleq2 2901	. . . . . 6 ⊢ (𝐴 = 𝐷 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐷))
23	22	anbi1d 631	. . . . 5 ⊢ (𝐴 = 𝐷 → ((𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐸 ∧ 𝑧 = 𝐹)) ↔ (𝑥 ∈ 𝐷 ∧ (𝑦 ∈ 𝐸 ∧ 𝑧 = 𝐹))))
24	21, 23	sylan9bbr 513	. . . 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 471	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 = 𝐶)))
26		anass 471	. . . 4 ⊢ (((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸) ∧ 𝑧 = 𝐹) ↔ (𝑥 ∈ 𝐷 ∧ (𝑦 ∈ 𝐸 ∧ 𝑧 = 𝐹)))
27	24, 25, 26	3bitr4g 316	. . 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 7219	. 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 7161	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
30		df-mpo 7161	. 2 ⊢ (𝑥 ∈ 𝐷, 𝑦 ∈ 𝐸 ↦ 𝐹) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸) ∧ 𝑧 = 𝐹)}
31	28, 29, 30	3eqtr4g 2881	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   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3138  {coprab 7157   ∈ cmpo 7158

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-oprab 7160  df-mpo 7161

This theorem is referenced by:  mpoeq12  7227  mapxpen  8683  pmatcollpw2lem  21385  xkoptsub  22262  xkocnv  22422  matunitlindflem1  34903