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Theorem abrexco 5426
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 2329 . . . . 5 (∃𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵}𝑥 = 𝐶 ↔ ∃𝑦(𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵} ∧ 𝑥 = 𝐶))
2 vex 2577 . . . . . . . . 9 𝑦 ∈ V
3 eqeq1 2062 . . . . . . . . . 10 (𝑧 = 𝑦 → (𝑧 = 𝐵𝑦 = 𝐵))
43rexbidv 2344 . . . . . . . . 9 (𝑧 = 𝑦 → (∃𝑤𝐴 𝑧 = 𝐵 ↔ ∃𝑤𝐴 𝑦 = 𝐵))
52, 4elab 2710 . . . . . . . 8 (𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵} ↔ ∃𝑤𝐴 𝑦 = 𝐵)
65anbi1i 439 . . . . . . 7 ((𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵} ∧ 𝑥 = 𝐶) ↔ (∃𝑤𝐴 𝑦 = 𝐵𝑥 = 𝐶))
7 r19.41v 2483 . . . . . . 7 (∃𝑤𝐴 (𝑦 = 𝐵𝑥 = 𝐶) ↔ (∃𝑤𝐴 𝑦 = 𝐵𝑥 = 𝐶))
86, 7bitr4i 180 . . . . . 6 ((𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵} ∧ 𝑥 = 𝐶) ↔ ∃𝑤𝐴 (𝑦 = 𝐵𝑥 = 𝐶))
98exbii 1512 . . . . 5 (∃𝑦(𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵} ∧ 𝑥 = 𝐶) ↔ ∃𝑦𝑤𝐴 (𝑦 = 𝐵𝑥 = 𝐶))
101, 9bitri 177 . . . 4 (∃𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵}𝑥 = 𝐶 ↔ ∃𝑦𝑤𝐴 (𝑦 = 𝐵𝑥 = 𝐶))
11 rexcom4 2594 . . . 4 (∃𝑤𝐴𝑦(𝑦 = 𝐵𝑥 = 𝐶) ↔ ∃𝑦𝑤𝐴 (𝑦 = 𝐵𝑥 = 𝐶))
1210, 11bitr4i 180 . . 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 2067 . . . . 5 (𝑦 = 𝐵 → (𝑥 = 𝐶𝑥 = 𝐷))
1613, 15ceqsexv 2610 . . . 4 (∃𝑦(𝑦 = 𝐵𝑥 = 𝐶) ↔ 𝑥 = 𝐷)
1716rexbii 2348 . . 3 (∃𝑤𝐴𝑦(𝑦 = 𝐵𝑥 = 𝐶) ↔ ∃𝑤𝐴 𝑥 = 𝐷)
1812, 17bitri 177 . 2 (∃𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵}𝑥 = 𝐶 ↔ ∃𝑤𝐴 𝑥 = 𝐷)
1918abbii 2169 1 {𝑥 ∣ ∃𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵}𝑥 = 𝐶} = {𝑥 ∣ ∃𝑤𝐴 𝑥 = 𝐷}
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wex 1397  wcel 1409  {cab 2042  wrex 2324  Vcvv 2574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576
This theorem is referenced by: (None)
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