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Theorem mpoeq123 7219
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 1916 . . . 4 𝑥 𝐴 = 𝐷
2 nfra1 3213 . . . 4 𝑥𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹)
31, 2nfan 1901 . . 3 𝑥(𝐴 = 𝐷 ∧ ∀𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹))
4 nfv 1916 . . . 4 𝑦 𝐴 = 𝐷
5 nfcv 2982 . . . . 5 𝑦𝐴
6 nfv 1916 . . . . . 6 𝑦 𝐵 = 𝐸
7 nfra1 3213 . . . . . 6 𝑦𝑦𝐵 𝐶 = 𝐹
86, 7nfan 1901 . . . . 5 𝑦(𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹)
95, 8nfralw 3219 . . . 4 𝑦𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹)
104, 9nfan 1901 . . 3 𝑦(𝐴 = 𝐷 ∧ ∀𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹))
11 nfv 1916 . . 3 𝑧(𝐴 = 𝐷 ∧ ∀𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹))
12 rsp 3200 . . . . . . 7 (∀𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹) → (𝑥𝐴 → (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹)))
13 rsp 3200 . . . . . . . . . 10 (∀𝑦𝐵 𝐶 = 𝐹 → (𝑦𝐵𝐶 = 𝐹))
14 eqeq2 2836 . . . . . . . . . 10 (𝐶 = 𝐹 → (𝑧 = 𝐶𝑧 = 𝐹))
1513, 14syl6 35 . . . . . . . . 9 (∀𝑦𝐵 𝐶 = 𝐹 → (𝑦𝐵 → (𝑧 = 𝐶𝑧 = 𝐹)))
1615pm5.32d 580 . . . . . . . 8 (∀𝑦𝐵 𝐶 = 𝐹 → ((𝑦𝐵𝑧 = 𝐶) ↔ (𝑦𝐵𝑧 = 𝐹)))
17 eleq2 2904 . . . . . . . . 9 (𝐵 = 𝐸 → (𝑦𝐵𝑦𝐸))
1817anbi1d 632 . . . . . . . 8 (𝐵 = 𝐸 → ((𝑦𝐵𝑧 = 𝐹) ↔ (𝑦𝐸𝑧 = 𝐹)))
1916, 18sylan9bbr 514 . . . . . . 7 ((𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹) → ((𝑦𝐵𝑧 = 𝐶) ↔ (𝑦𝐸𝑧 = 𝐹)))
2012, 19syl6 35 . . . . . 6 (∀𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹) → (𝑥𝐴 → ((𝑦𝐵𝑧 = 𝐶) ↔ (𝑦𝐸𝑧 = 𝐹))))
2120pm5.32d 580 . . . . 5 (∀𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹) → ((𝑥𝐴 ∧ (𝑦𝐵𝑧 = 𝐶)) ↔ (𝑥𝐴 ∧ (𝑦𝐸𝑧 = 𝐹))))
22 eleq2 2904 . . . . . 6 (𝐴 = 𝐷 → (𝑥𝐴𝑥𝐷))
2322anbi1d 632 . . . . 5 (𝐴 = 𝐷 → ((𝑥𝐴 ∧ (𝑦𝐸𝑧 = 𝐹)) ↔ (𝑥𝐷 ∧ (𝑦𝐸𝑧 = 𝐹))))
2421, 23sylan9bbr 514 . . . 4 ((𝐴 = 𝐷 ∧ ∀𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹)) → ((𝑥𝐴 ∧ (𝑦𝐵𝑧 = 𝐶)) ↔ (𝑥𝐷 ∧ (𝑦𝐸𝑧 = 𝐹))))
25 anass 472 . . . 4 (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ (𝑥𝐴 ∧ (𝑦𝐵𝑧 = 𝐶)))
26 anass 472 . . . 4 (((𝑥𝐷𝑦𝐸) ∧ 𝑧 = 𝐹) ↔ (𝑥𝐷 ∧ (𝑦𝐸𝑧 = 𝐹)))
2724, 25, 263bitr4g 317 . . 3 ((𝐴 = 𝐷 ∧ ∀𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹)) → (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥𝐷𝑦𝐸) ∧ 𝑧 = 𝐹)))
283, 10, 11, 27oprabbid 7212 . 2 ((𝐴 = 𝐷 ∧ ∀𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹)) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐷𝑦𝐸) ∧ 𝑧 = 𝐹)})
29 df-mpo 7154 . 2 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
30 df-mpo 7154 . 2 (𝑥𝐷, 𝑦𝐸𝐹) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐷𝑦𝐸) ∧ 𝑧 = 𝐹)}
3128, 29, 303eqtr4g 2884 1 ((𝐴 = 𝐷 ∧ ∀𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹)) → (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐷, 𝑦𝐸𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wral 3133  {coprab 7150  cmpo 7151
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-oprab 7153  df-mpo 7154
This theorem is referenced by:  mpoeq12  7220  mapxpen  8680  pmatcollpw2lem  21388  xkoptsub  22265  xkocnv  22425  matunitlindflem1  35001
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