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

Theorem abrexco 5319
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 2286 . . . . 5 (y {zw A z = B}x = 𝐶y(y {zw A z = B} x = 𝐶))
2 vex 2534 . . . . . . . . 9 y V
3 eqeq1 2024 . . . . . . . . . 10 (z = y → (z = By = B))
43rexbidv 2301 . . . . . . . . 9 (z = y → (w A z = Bw A y = B))
52, 4elab 2660 . . . . . . . 8 (y {zw A z = B} ↔ w A y = B)
65anbi1i 434 . . . . . . 7 ((y {zw A z = B} x = 𝐶) ↔ (w A y = B x = 𝐶))
7 r19.41v 2440 . . . . . . 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 1474 . . . . 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 2550 . . . 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 2029 . . . . 5 (y = B → (x = 𝐶x = 𝐷))
1613, 15ceqsexv 2566 . . . 4 (y(y = B x = 𝐶) ↔ x = 𝐷)
1716rexbii 2305 . . 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 2131 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 1226  wex 1358   wcel 1370  {cab 2004  wrex 2281  Vcvv 2531
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-rex 2286  df-v 2533
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator