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Theorem mpt2eq123 6948
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 2010 . . . 4 𝑥 𝐴 = 𝐷
2 nfra1 3122 . . . 4 𝑥𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹)
31, 2nfan 1999 . . 3 𝑥(𝐴 = 𝐷 ∧ ∀𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹))
4 nfv 2010 . . . 4 𝑦 𝐴 = 𝐷
5 nfcv 2941 . . . . 5 𝑦𝐴
6 nfv 2010 . . . . . 6 𝑦 𝐵 = 𝐸
7 nfra1 3122 . . . . . 6 𝑦𝑦𝐵 𝐶 = 𝐹
86, 7nfan 1999 . . . . 5 𝑦(𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹)
95, 8nfral 3126 . . . 4 𝑦𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹)
104, 9nfan 1999 . . 3 𝑦(𝐴 = 𝐷 ∧ ∀𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹))
11 nfv 2010 . . 3 𝑧(𝐴 = 𝐷 ∧ ∀𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹))
12 rsp 3110 . . . . . . 7 (∀𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹) → (𝑥𝐴 → (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹)))
13 rsp 3110 . . . . . . . . . 10 (∀𝑦𝐵 𝐶 = 𝐹 → (𝑦𝐵𝐶 = 𝐹))
14 eqeq2 2810 . . . . . . . . . 10 (𝐶 = 𝐹 → (𝑧 = 𝐶𝑧 = 𝐹))
1513, 14syl6 35 . . . . . . . . 9 (∀𝑦𝐵 𝐶 = 𝐹 → (𝑦𝐵 → (𝑧 = 𝐶𝑧 = 𝐹)))
1615pm5.32d 573 . . . . . . . 8 (∀𝑦𝐵 𝐶 = 𝐹 → ((𝑦𝐵𝑧 = 𝐶) ↔ (𝑦𝐵𝑧 = 𝐹)))
17 eleq2 2867 . . . . . . . . 9 (𝐵 = 𝐸 → (𝑦𝐵𝑦𝐸))
1817anbi1d 624 . . . . . . . 8 (𝐵 = 𝐸 → ((𝑦𝐵𝑧 = 𝐹) ↔ (𝑦𝐸𝑧 = 𝐹)))
1916, 18sylan9bbr 507 . . . . . . 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 573 . . . . 5 (∀𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹) → ((𝑥𝐴 ∧ (𝑦𝐵𝑧 = 𝐶)) ↔ (𝑥𝐴 ∧ (𝑦𝐸𝑧 = 𝐹))))
22 eleq2 2867 . . . . . 6 (𝐴 = 𝐷 → (𝑥𝐴𝑥𝐷))
2322anbi1d 624 . . . . 5 (𝐴 = 𝐷 → ((𝑥𝐴 ∧ (𝑦𝐸𝑧 = 𝐹)) ↔ (𝑥𝐷 ∧ (𝑦𝐸𝑧 = 𝐹))))
2421, 23sylan9bbr 507 . . . 4 ((𝐴 = 𝐷 ∧ ∀𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹)) → ((𝑥𝐴 ∧ (𝑦𝐵𝑧 = 𝐶)) ↔ (𝑥𝐷 ∧ (𝑦𝐸𝑧 = 𝐹))))
25 anass 461 . . . 4 (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ (𝑥𝐴 ∧ (𝑦𝐵𝑧 = 𝐶)))
26 anass 461 . . . 4 (((𝑥𝐷𝑦𝐸) ∧ 𝑧 = 𝐹) ↔ (𝑥𝐷 ∧ (𝑦𝐸𝑧 = 𝐹)))
2724, 25, 263bitr4g 306 . . 3 ((𝐴 = 𝐷 ∧ ∀𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹)) → (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥𝐷𝑦𝐸) ∧ 𝑧 = 𝐹)))
283, 10, 11, 27oprabbid 6942 . 2 ((𝐴 = 𝐷 ∧ ∀𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹)) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐷𝑦𝐸) ∧ 𝑧 = 𝐹)})
29 df-mpt2 6883 . 2 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
30 df-mpt2 6883 . 2 (𝑥𝐷, 𝑦𝐸𝐹) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐷𝑦𝐸) ∧ 𝑧 = 𝐹)}
3128, 29, 303eqtr4g 2858 1 ((𝐴 = 𝐷 ∧ ∀𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹)) → (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐷, 𝑦𝐸𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  wral 3089  {coprab 6879  cmpt2 6880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-oprab 6882  df-mpt2 6883
This theorem is referenced by:  mpt2eq12  6949  mapxpen  8368  pmatcollpw2lem  20910  xkoptsub  21786  xkocnv  21946  matunitlindflem1  33894
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