ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  mpt2eq123 Structured version   GIF version

Theorem mpt2eq123 5487
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 x A (B = 𝐸 y B 𝐶 = 𝐹)) → (x A, y B𝐶) = (x 𝐷, y 𝐸𝐹))
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 1402 . . . 4 x A = 𝐷
2 nfra1 2333 . . . 4 xx A (B = 𝐸 y B 𝐶 = 𝐹)
31, 2nfan 1439 . . 3 x(A = 𝐷 x A (B = 𝐸 y B 𝐶 = 𝐹))
4 nfv 1402 . . . 4 y A = 𝐷
5 nfcv 2160 . . . . 5 yA
6 nfv 1402 . . . . . 6 y B = 𝐸
7 nfra1 2333 . . . . . 6 yy B 𝐶 = 𝐹
86, 7nfan 1439 . . . . 5 y(B = 𝐸 y B 𝐶 = 𝐹)
95, 8nfralxy 2338 . . . 4 yx A (B = 𝐸 y B 𝐶 = 𝐹)
104, 9nfan 1439 . . 3 y(A = 𝐷 x A (B = 𝐸 y B 𝐶 = 𝐹))
11 nfv 1402 . . 3 z(A = 𝐷 x A (B = 𝐸 y B 𝐶 = 𝐹))
12 rsp 2347 . . . . . . 7 (x A (B = 𝐸 y B 𝐶 = 𝐹) → (x A → (B = 𝐸 y B 𝐶 = 𝐹)))
13 rsp 2347 . . . . . . . . . 10 (y B 𝐶 = 𝐹 → (y B𝐶 = 𝐹))
14 eqeq2 2031 . . . . . . . . . 10 (𝐶 = 𝐹 → (z = 𝐶z = 𝐹))
1513, 14syl6 29 . . . . . . . . 9 (y B 𝐶 = 𝐹 → (y B → (z = 𝐶z = 𝐹)))
1615pm5.32d 426 . . . . . . . 8 (y B 𝐶 = 𝐹 → ((y B z = 𝐶) ↔ (y B z = 𝐹)))
17 eleq2 2083 . . . . . . . . 9 (B = 𝐸 → (y By 𝐸))
1817anbi1d 441 . . . . . . . 8 (B = 𝐸 → ((y B z = 𝐹) ↔ (y 𝐸 z = 𝐹)))
1916, 18sylan9bbr 439 . . . . . . 7 ((B = 𝐸 y B 𝐶 = 𝐹) → ((y B z = 𝐶) ↔ (y 𝐸 z = 𝐹)))
2012, 19syl6 29 . . . . . 6 (x A (B = 𝐸 y B 𝐶 = 𝐹) → (x A → ((y B z = 𝐶) ↔ (y 𝐸 z = 𝐹))))
2120pm5.32d 426 . . . . 5 (x A (B = 𝐸 y B 𝐶 = 𝐹) → ((x A (y B z = 𝐶)) ↔ (x A (y 𝐸 z = 𝐹))))
22 eleq2 2083 . . . . . 6 (A = 𝐷 → (x Ax 𝐷))
2322anbi1d 441 . . . . 5 (A = 𝐷 → ((x A (y 𝐸 z = 𝐹)) ↔ (x 𝐷 (y 𝐸 z = 𝐹))))
2421, 23sylan9bbr 439 . . . 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 383 . . . 4 (((x A y B) z = 𝐶) ↔ (x A (y B z = 𝐶)))
26 anass 383 . . . 4 (((x 𝐷 y 𝐸) z = 𝐹) ↔ (x 𝐷 (y 𝐸 z = 𝐹)))
2724, 25, 263bitr4g 212 . . 3 ((A = 𝐷 x A (B = 𝐸 y B 𝐶 = 𝐹)) → (((x A y B) z = 𝐶) ↔ ((x 𝐷 y 𝐸) z = 𝐹)))
283, 10, 11, 27oprabbid 5481 . 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-mpt2 5441 . 2 (x A, y B𝐶) = {⟨⟨x, y⟩, z⟩ ∣ ((x A y B) z = 𝐶)}
30 df-mpt2 5441 . 2 (x 𝐷, y 𝐸𝐹) = {⟨⟨x, y⟩, z⟩ ∣ ((x 𝐷 y 𝐸) z = 𝐹)}
3128, 29, 303eqtr4g 2079 1 ((A = 𝐷 x A (B = 𝐸 y B 𝐶 = 𝐹)) → (x A, y B𝐶) = (x 𝐷, y 𝐸𝐹))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1228   wcel 1374  wral 2284  {coprab 5437  cmpt2 5438
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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-11 1378  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-oprab 5440  df-mpt2 5441
This theorem is referenced by:  mpt2eq12  5488
  Copyright terms: Public domain W3C validator