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Theorem mpt2eq123 5503
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 1418 . . . 4 x A = 𝐷
2 nfra1 2349 . . . 4 xx A (B = 𝐸 y B 𝐶 = 𝐹)
31, 2nfan 1454 . . 3 x(A = 𝐷 x A (B = 𝐸 y B 𝐶 = 𝐹))
4 nfv 1418 . . . 4 y A = 𝐷
5 nfcv 2175 . . . . 5 yA
6 nfv 1418 . . . . . 6 y B = 𝐸
7 nfra1 2349 . . . . . 6 yy B 𝐶 = 𝐹
86, 7nfan 1454 . . . . 5 y(B = 𝐸 y B 𝐶 = 𝐹)
95, 8nfralxy 2354 . . . 4 yx A (B = 𝐸 y B 𝐶 = 𝐹)
104, 9nfan 1454 . . 3 y(A = 𝐷 x A (B = 𝐸 y B 𝐶 = 𝐹))
11 nfv 1418 . . 3 z(A = 𝐷 x A (B = 𝐸 y B 𝐶 = 𝐹))
12 rsp 2363 . . . . . . 7 (x A (B = 𝐸 y B 𝐶 = 𝐹) → (x A → (B = 𝐸 y B 𝐶 = 𝐹)))
13 rsp 2363 . . . . . . . . . 10 (y B 𝐶 = 𝐹 → (y B𝐶 = 𝐹))
14 eqeq2 2046 . . . . . . . . . 10 (𝐶 = 𝐹 → (z = 𝐶z = 𝐹))
1513, 14syl6 29 . . . . . . . . 9 (y B 𝐶 = 𝐹 → (y B → (z = 𝐶z = 𝐹)))
1615pm5.32d 423 . . . . . . . 8 (y B 𝐶 = 𝐹 → ((y B z = 𝐶) ↔ (y B z = 𝐹)))
17 eleq2 2098 . . . . . . . . 9 (B = 𝐸 → (y By 𝐸))
1817anbi1d 438 . . . . . . . 8 (B = 𝐸 → ((y B z = 𝐹) ↔ (y 𝐸 z = 𝐹)))
1916, 18sylan9bbr 436 . . . . . . 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 423 . . . . 5 (x A (B = 𝐸 y B 𝐶 = 𝐹) → ((x A (y B z = 𝐶)) ↔ (x A (y 𝐸 z = 𝐹))))
22 eleq2 2098 . . . . . 6 (A = 𝐷 → (x Ax 𝐷))
2322anbi1d 438 . . . . 5 (A = 𝐷 → ((x A (y 𝐸 z = 𝐹)) ↔ (x 𝐷 (y 𝐸 z = 𝐹))))
2421, 23sylan9bbr 436 . . . 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 381 . . . 4 (((x A y B) z = 𝐶) ↔ (x A (y B z = 𝐶)))
26 anass 381 . . . 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 5497 . 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 5457 . 2 (x A, y B𝐶) = {⟨⟨x, y⟩, z⟩ ∣ ((x A y B) z = 𝐶)}
30 df-mpt2 5457 . 2 (x 𝐷, y 𝐸𝐹) = {⟨⟨x, y⟩, z⟩ ∣ ((x 𝐷 y 𝐸) z = 𝐹)}
3128, 29, 303eqtr4g 2094 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 1242   wcel 1390  wral 2300  {coprab 5453  cmpt2 5454
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 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-oprab 5456  df-mpt2 5457
This theorem is referenced by:  mpt2eq12  5504
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