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Theorem mpoeq123 5830
 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 ∧ ∀𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹)) → (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐷, 𝑦𝐸𝐹))
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.
StepHypRef Expression
1 nfv 1508 . . . 4 𝑥 𝐴 = 𝐷
2 nfra1 2466 . . . 4 𝑥𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹)
31, 2nfan 1544 . . 3 𝑥(𝐴 = 𝐷 ∧ ∀𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹))
4 nfv 1508 . . . 4 𝑦 𝐴 = 𝐷
5 nfcv 2281 . . . . 5 𝑦𝐴
6 nfv 1508 . . . . . 6 𝑦 𝐵 = 𝐸
7 nfra1 2466 . . . . . 6 𝑦𝑦𝐵 𝐶 = 𝐹
86, 7nfan 1544 . . . . 5 𝑦(𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹)
95, 8nfralxy 2471 . . . 4 𝑦𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹)
104, 9nfan 1544 . . 3 𝑦(𝐴 = 𝐷 ∧ ∀𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹))
11 nfv 1508 . . 3 𝑧(𝐴 = 𝐷 ∧ ∀𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹))
12 rsp 2480 . . . . . . 7 (∀𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹) → (𝑥𝐴 → (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹)))
13 rsp 2480 . . . . . . . . . 10 (∀𝑦𝐵 𝐶 = 𝐹 → (𝑦𝐵𝐶 = 𝐹))
14 eqeq2 2149 . . . . . . . . . 10 (𝐶 = 𝐹 → (𝑧 = 𝐶𝑧 = 𝐹))
1513, 14syl6 33 . . . . . . . . 9 (∀𝑦𝐵 𝐶 = 𝐹 → (𝑦𝐵 → (𝑧 = 𝐶𝑧 = 𝐹)))
1615pm5.32d 445 . . . . . . . 8 (∀𝑦𝐵 𝐶 = 𝐹 → ((𝑦𝐵𝑧 = 𝐶) ↔ (𝑦𝐵𝑧 = 𝐹)))
17 eleq2 2203 . . . . . . . . 9 (𝐵 = 𝐸 → (𝑦𝐵𝑦𝐸))
1817anbi1d 460 . . . . . . . 8 (𝐵 = 𝐸 → ((𝑦𝐵𝑧 = 𝐹) ↔ (𝑦𝐸𝑧 = 𝐹)))
1916, 18sylan9bbr 458 . . . . . . 7 ((𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹) → ((𝑦𝐵𝑧 = 𝐶) ↔ (𝑦𝐸𝑧 = 𝐹)))
2012, 19syl6 33 . . . . . 6 (∀𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹) → (𝑥𝐴 → ((𝑦𝐵𝑧 = 𝐶) ↔ (𝑦𝐸𝑧 = 𝐹))))
2120pm5.32d 445 . . . . 5 (∀𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹) → ((𝑥𝐴 ∧ (𝑦𝐵𝑧 = 𝐶)) ↔ (𝑥𝐴 ∧ (𝑦𝐸𝑧 = 𝐹))))
22 eleq2 2203 . . . . . 6 (𝐴 = 𝐷 → (𝑥𝐴𝑥𝐷))
2322anbi1d 460 . . . . 5 (𝐴 = 𝐷 → ((𝑥𝐴 ∧ (𝑦𝐸𝑧 = 𝐹)) ↔ (𝑥𝐷 ∧ (𝑦𝐸𝑧 = 𝐹))))
2421, 23sylan9bbr 458 . . . 4 ((𝐴 = 𝐷 ∧ ∀𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹)) → ((𝑥𝐴 ∧ (𝑦𝐵𝑧 = 𝐶)) ↔ (𝑥𝐷 ∧ (𝑦𝐸𝑧 = 𝐹))))
25 anass 398 . . . 4 (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ (𝑥𝐴 ∧ (𝑦𝐵𝑧 = 𝐶)))
26 anass 398 . . . 4 (((𝑥𝐷𝑦𝐸) ∧ 𝑧 = 𝐹) ↔ (𝑥𝐷 ∧ (𝑦𝐸𝑧 = 𝐹)))
2724, 25, 263bitr4g 222 . . 3 ((𝐴 = 𝐷 ∧ ∀𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹)) → (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥𝐷𝑦𝐸) ∧ 𝑧 = 𝐹)))
283, 10, 11, 27oprabbid 5824 . 2 ((𝐴 = 𝐷 ∧ ∀𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹)) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐷𝑦𝐸) ∧ 𝑧 = 𝐹)})
29 df-mpo 5779 . 2 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
30 df-mpo 5779 . 2 (𝑥𝐷, 𝑦𝐸𝐹) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐷𝑦𝐸) ∧ 𝑧 = 𝐹)}
3128, 29, 303eqtr4g 2197 1 ((𝐴 = 𝐷 ∧ ∀𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹)) → (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐷, 𝑦𝐸𝐹))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331   ∈ wcel 1480  ∀wral 2416  {coprab 5775   ∈ cmpo 5776 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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-oprab 5778  df-mpo 5779 This theorem is referenced by:  mpoeq12  5831  mapxpen  6742
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