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Theorem mpt2eq123 5564
 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
mpt2eq123 ((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 mpt2eq123
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfv 1421 . . . 4 𝑥 𝐴 = 𝐷
2 nfra1 2355 . . . 4 𝑥𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹)
31, 2nfan 1457 . . 3 𝑥(𝐴 = 𝐷 ∧ ∀𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹))
4 nfv 1421 . . . 4 𝑦 𝐴 = 𝐷
5 nfcv 2178 . . . . 5 𝑦𝐴
6 nfv 1421 . . . . . 6 𝑦 𝐵 = 𝐸
7 nfra1 2355 . . . . . 6 𝑦𝑦𝐵 𝐶 = 𝐹
86, 7nfan 1457 . . . . 5 𝑦(𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹)
95, 8nfralxy 2360 . . . 4 𝑦𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹)
104, 9nfan 1457 . . 3 𝑦(𝐴 = 𝐷 ∧ ∀𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹))
11 nfv 1421 . . 3 𝑧(𝐴 = 𝐷 ∧ ∀𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹))
12 rsp 2369 . . . . . . 7 (∀𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹) → (𝑥𝐴 → (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹)))
13 rsp 2369 . . . . . . . . . 10 (∀𝑦𝐵 𝐶 = 𝐹 → (𝑦𝐵𝐶 = 𝐹))
14 eqeq2 2049 . . . . . . . . . 10 (𝐶 = 𝐹 → (𝑧 = 𝐶𝑧 = 𝐹))
1513, 14syl6 29 . . . . . . . . 9 (∀𝑦𝐵 𝐶 = 𝐹 → (𝑦𝐵 → (𝑧 = 𝐶𝑧 = 𝐹)))
1615pm5.32d 423 . . . . . . . 8 (∀𝑦𝐵 𝐶 = 𝐹 → ((𝑦𝐵𝑧 = 𝐶) ↔ (𝑦𝐵𝑧 = 𝐹)))
17 eleq2 2101 . . . . . . . . 9 (𝐵 = 𝐸 → (𝑦𝐵𝑦𝐸))
1817anbi1d 438 . . . . . . . 8 (𝐵 = 𝐸 → ((𝑦𝐵𝑧 = 𝐹) ↔ (𝑦𝐸𝑧 = 𝐹)))
1916, 18sylan9bbr 436 . . . . . . 7 ((𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹) → ((𝑦𝐵𝑧 = 𝐶) ↔ (𝑦𝐸𝑧 = 𝐹)))
2012, 19syl6 29 . . . . . 6 (∀𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹) → (𝑥𝐴 → ((𝑦𝐵𝑧 = 𝐶) ↔ (𝑦𝐸𝑧 = 𝐹))))
2120pm5.32d 423 . . . . 5 (∀𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹) → ((𝑥𝐴 ∧ (𝑦𝐵𝑧 = 𝐶)) ↔ (𝑥𝐴 ∧ (𝑦𝐸𝑧 = 𝐹))))
22 eleq2 2101 . . . . . 6 (𝐴 = 𝐷 → (𝑥𝐴𝑥𝐷))
2322anbi1d 438 . . . . 5 (𝐴 = 𝐷 → ((𝑥𝐴 ∧ (𝑦𝐸𝑧 = 𝐹)) ↔ (𝑥𝐷 ∧ (𝑦𝐸𝑧 = 𝐹))))
2421, 23sylan9bbr 436 . . . 4 ((𝐴 = 𝐷 ∧ ∀𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹)) → ((𝑥𝐴 ∧ (𝑦𝐵𝑧 = 𝐶)) ↔ (𝑥𝐷 ∧ (𝑦𝐸𝑧 = 𝐹))))
25 anass 381 . . . 4 (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ (𝑥𝐴 ∧ (𝑦𝐵𝑧 = 𝐶)))
26 anass 381 . . . 4 (((𝑥𝐷𝑦𝐸) ∧ 𝑧 = 𝐹) ↔ (𝑥𝐷 ∧ (𝑦𝐸𝑧 = 𝐹)))
2724, 25, 263bitr4g 212 . . 3 ((𝐴 = 𝐷 ∧ ∀𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹)) → (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥𝐷𝑦𝐸) ∧ 𝑧 = 𝐹)))
283, 10, 11, 27oprabbid 5558 . 2 ((𝐴 = 𝐷 ∧ ∀𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹)) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐷𝑦𝐸) ∧ 𝑧 = 𝐹)})
29 df-mpt2 5517 . 2 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
30 df-mpt2 5517 . 2 (𝑥𝐷, 𝑦𝐸𝐹) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐷𝑦𝐸) ∧ 𝑧 = 𝐹)}
3128, 29, 303eqtr4g 2097 1 ((𝐴 = 𝐷 ∧ ∀𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹)) → (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐷, 𝑦𝐸𝐹))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243   ∈ wcel 1393  ∀wral 2306  {coprab 5513   ↦ cmpt2 5514 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-oprab 5516  df-mpt2 5517 This theorem is referenced by:  mpt2eq12  5565
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