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Theorem abrexco 5653
 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 {𝑥 ∣ ∃𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵}𝑥 = 𝐶} = {𝑥 ∣ ∃𝑤𝐴 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 2420 . . . . 5 (∃𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵}𝑥 = 𝐶 ↔ ∃𝑦(𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵} ∧ 𝑥 = 𝐶))
2 vex 2684 . . . . . . . . 9 𝑦 ∈ V
3 eqeq1 2144 . . . . . . . . . 10 (𝑧 = 𝑦 → (𝑧 = 𝐵𝑦 = 𝐵))
43rexbidv 2436 . . . . . . . . 9 (𝑧 = 𝑦 → (∃𝑤𝐴 𝑧 = 𝐵 ↔ ∃𝑤𝐴 𝑦 = 𝐵))
52, 4elab 2823 . . . . . . . 8 (𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵} ↔ ∃𝑤𝐴 𝑦 = 𝐵)
65anbi1i 453 . . . . . . 7 ((𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵} ∧ 𝑥 = 𝐶) ↔ (∃𝑤𝐴 𝑦 = 𝐵𝑥 = 𝐶))
7 r19.41v 2585 . . . . . . 7 (∃𝑤𝐴 (𝑦 = 𝐵𝑥 = 𝐶) ↔ (∃𝑤𝐴 𝑦 = 𝐵𝑥 = 𝐶))
86, 7bitr4i 186 . . . . . 6 ((𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵} ∧ 𝑥 = 𝐶) ↔ ∃𝑤𝐴 (𝑦 = 𝐵𝑥 = 𝐶))
98exbii 1584 . . . . 5 (∃𝑦(𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵} ∧ 𝑥 = 𝐶) ↔ ∃𝑦𝑤𝐴 (𝑦 = 𝐵𝑥 = 𝐶))
101, 9bitri 183 . . . 4 (∃𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵}𝑥 = 𝐶 ↔ ∃𝑦𝑤𝐴 (𝑦 = 𝐵𝑥 = 𝐶))
11 rexcom4 2704 . . . 4 (∃𝑤𝐴𝑦(𝑦 = 𝐵𝑥 = 𝐶) ↔ ∃𝑦𝑤𝐴 (𝑦 = 𝐵𝑥 = 𝐶))
1210, 11bitr4i 186 . . 3 (∃𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵}𝑥 = 𝐶 ↔ ∃𝑤𝐴𝑦(𝑦 = 𝐵𝑥 = 𝐶))
13 abrexco.1 . . . . 5 𝐵 ∈ V
14 abrexco.2 . . . . . 6 (𝑦 = 𝐵𝐶 = 𝐷)
1514eqeq2d 2149 . . . . 5 (𝑦 = 𝐵 → (𝑥 = 𝐶𝑥 = 𝐷))
1613, 15ceqsexv 2720 . . . 4 (∃𝑦(𝑦 = 𝐵𝑥 = 𝐶) ↔ 𝑥 = 𝐷)
1716rexbii 2440 . . 3 (∃𝑤𝐴𝑦(𝑦 = 𝐵𝑥 = 𝐶) ↔ ∃𝑤𝐴 𝑥 = 𝐷)
1812, 17bitri 183 . 2 (∃𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵}𝑥 = 𝐶 ↔ ∃𝑤𝐴 𝑥 = 𝐷)
1918abbii 2253 1 {𝑥 ∣ ∃𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵}𝑥 = 𝐶} = {𝑥 ∣ ∃𝑤𝐴 𝑥 = 𝐷}
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331  ∃wex 1468   ∈ wcel 1480  {cab 2123  ∃wrex 2415  Vcvv 2681 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-v 2683 This theorem is referenced by:  restco  12332
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