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Theorem mpt2eq123 5661
 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 B C) = (x D, y E F))
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 1619 . . . 4 x A = D
2 nfra1 2664 . . . 4 xx A (B = E y B C = F)
31, 2nfan 1824 . . 3 x(A = D x A (B = E y B C = F))
4 nfv 1619 . . . 4 y A = D
5 nfcv 2489 . . . . 5 yA
6 nfv 1619 . . . . . 6 y B = E
7 nfra1 2664 . . . . . 6 yy B C = F
86, 7nfan 1824 . . . . 5 y(B = E y B C = F)
95, 8nfral 2667 . . . 4 yx A (B = E y B C = F)
104, 9nfan 1824 . . 3 y(A = D x A (B = E y B C = F))
11 nfv 1619 . . 3 z(A = D x A (B = E y B C = F))
12 rsp 2674 . . . . . . 7 (x A (B = E y B C = F) → (x A → (B = E y B C = F)))
13 rsp 2674 . . . . . . . . . 10 (y B C = F → (y BC = F))
14 eqeq2 2362 . . . . . . . . . 10 (C = F → (z = Cz = F))
1513, 14syl6 29 . . . . . . . . 9 (y B C = F → (y B → (z = Cz = F)))
1615pm5.32d 620 . . . . . . . 8 (y B C = F → ((y B z = C) ↔ (y B z = F)))
17 eleq2 2414 . . . . . . . . 9 (B = E → (y By E))
1817anbi1d 685 . . . . . . . 8 (B = E → ((y B z = F) ↔ (y E z = F)))
1916, 18sylan9bbr 681 . . . . . . 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 620 . . . . 5 (x A (B = E y B C = F) → ((x A (y B z = C)) ↔ (x A (y E z = F))))
22 eleq2 2414 . . . . . 6 (A = D → (x Ax D))
2322anbi1d 685 . . . . 5 (A = D → ((x A (y E z = F)) ↔ (x D (y E z = F))))
2421, 23sylan9bbr 681 . . . 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 630 . . . 4 (((x A y B) z = C) ↔ (x A (y B z = C)))
26 anass 630 . . . 4 (((x D y E) z = F) ↔ (x D (y E z = F)))
2724, 25, 263bitr4g 279 . . 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 5563 . 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 5654 . 2 (x A, y B C) = {x, y, z ((x A y B) z = C)}
30 df-mpt2 5654 . 2 (x D, y E F) = {x, y, z ((x D y E) z = F)}
3128, 29, 303eqtr4g 2410 1 ((A = D x A (B = E y B C = F)) → (x A, y B C) = (x D, y E F))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2614  {coprab 5527   ↦ cmpt2 5653 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-oprab 5528  df-mpt2 5654 This theorem is referenced by:  mpt2eq12  5662
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