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Theorem abrexco 5341
Description: Composition of two image maps 𝐶(y) and B(w). (Contributed by NM, 27-May-2013.)
Hypotheses
Ref Expression
abrexco.1 B V
abrexco.2 (y = B𝐶 = 𝐷)
Assertion
Ref Expression
abrexco {xy {zw A z = B}x = 𝐶} = {xw A x = 𝐷}
Distinct variable groups:   y,A,z   y,B,z   w,𝐶   y,𝐷   x,w,y   z,w
Allowed substitution hints:   A(x,w)   B(x,w)   𝐶(x,y,z)   𝐷(x,z,w)

Proof of Theorem abrexco
StepHypRef Expression
1 df-rex 2306 . . . . 5 (y {zw A z = B}x = 𝐶y(y {zw A z = B} x = 𝐶))
2 vex 2554 . . . . . . . . 9 y V
3 eqeq1 2043 . . . . . . . . . 10 (z = y → (z = By = B))
43rexbidv 2321 . . . . . . . . 9 (z = y → (w A z = Bw A y = B))
52, 4elab 2681 . . . . . . . 8 (y {zw A z = B} ↔ w A y = B)
65anbi1i 431 . . . . . . 7 ((y {zw A z = B} x = 𝐶) ↔ (w A y = B x = 𝐶))
7 r19.41v 2460 . . . . . . 7 (w A (y = B x = 𝐶) ↔ (w A y = B x = 𝐶))
86, 7bitr4i 176 . . . . . 6 ((y {zw A z = B} x = 𝐶) ↔ w A (y = B x = 𝐶))
98exbii 1493 . . . . 5 (y(y {zw A z = B} x = 𝐶) ↔ yw A (y = B x = 𝐶))
101, 9bitri 173 . . . 4 (y {zw A z = B}x = 𝐶yw A (y = B x = 𝐶))
11 rexcom4 2571 . . . 4 (w A y(y = B x = 𝐶) ↔ yw A (y = B x = 𝐶))
1210, 11bitr4i 176 . . 3 (y {zw A z = B}x = 𝐶w A y(y = B x = 𝐶))
13 abrexco.1 . . . . 5 B V
14 abrexco.2 . . . . . 6 (y = B𝐶 = 𝐷)
1514eqeq2d 2048 . . . . 5 (y = B → (x = 𝐶x = 𝐷))
1613, 15ceqsexv 2587 . . . 4 (y(y = B x = 𝐶) ↔ x = 𝐷)
1716rexbii 2325 . . 3 (w A y(y = B x = 𝐶) ↔ w A x = 𝐷)
1812, 17bitri 173 . 2 (y {zw A z = B}x = 𝐶w A x = 𝐷)
1918abbii 2150 1 {xy {zw A z = B}x = 𝐶} = {xw A x = 𝐷}
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242  wex 1378   wcel 1390  {cab 2023  wrex 2301  Vcvv 2551
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-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  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-rex 2306  df-v 2553
This theorem is referenced by: (None)
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