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Theorem abrexco 6985
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 3115 . . . . 5 (∃𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵}𝑥 = 𝐶 ↔ ∃𝑦(𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵} ∧ 𝑥 = 𝐶))
2 vex 3447 . . . . . . . . 9 𝑦 ∈ V
3 eqeq1 2805 . . . . . . . . . 10 (𝑧 = 𝑦 → (𝑧 = 𝐵𝑦 = 𝐵))
43rexbidv 3259 . . . . . . . . 9 (𝑧 = 𝑦 → (∃𝑤𝐴 𝑧 = 𝐵 ↔ ∃𝑤𝐴 𝑦 = 𝐵))
52, 4elab 3618 . . . . . . . 8 (𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵} ↔ ∃𝑤𝐴 𝑦 = 𝐵)
65anbi1i 626 . . . . . . 7 ((𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵} ∧ 𝑥 = 𝐶) ↔ (∃𝑤𝐴 𝑦 = 𝐵𝑥 = 𝐶))
7 r19.41v 3303 . . . . . . 7 (∃𝑤𝐴 (𝑦 = 𝐵𝑥 = 𝐶) ↔ (∃𝑤𝐴 𝑦 = 𝐵𝑥 = 𝐶))
86, 7bitr4i 281 . . . . . 6 ((𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵} ∧ 𝑥 = 𝐶) ↔ ∃𝑤𝐴 (𝑦 = 𝐵𝑥 = 𝐶))
98exbii 1849 . . . . 5 (∃𝑦(𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵} ∧ 𝑥 = 𝐶) ↔ ∃𝑦𝑤𝐴 (𝑦 = 𝐵𝑥 = 𝐶))
101, 9bitri 278 . . . 4 (∃𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵}𝑥 = 𝐶 ↔ ∃𝑦𝑤𝐴 (𝑦 = 𝐵𝑥 = 𝐶))
11 rexcom4 3215 . . . 4 (∃𝑤𝐴𝑦(𝑦 = 𝐵𝑥 = 𝐶) ↔ ∃𝑦𝑤𝐴 (𝑦 = 𝐵𝑥 = 𝐶))
1210, 11bitr4i 281 . . 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 2812 . . . . 5 (𝑦 = 𝐵 → (𝑥 = 𝐶𝑥 = 𝐷))
1613, 15ceqsexv 3492 . . . 4 (∃𝑦(𝑦 = 𝐵𝑥 = 𝐶) ↔ 𝑥 = 𝐷)
1716rexbii 3213 . . 3 (∃𝑤𝐴𝑦(𝑦 = 𝐵𝑥 = 𝐶) ↔ ∃𝑤𝐴 𝑥 = 𝐷)
1812, 17bitri 278 . 2 (∃𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵}𝑥 = 𝐶 ↔ ∃𝑤𝐴 𝑥 = 𝐷)
1918abbii 2866 1 {𝑥 ∣ ∃𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵}𝑥 = 𝐶} = {𝑥 ∣ ∃𝑤𝐴 𝑥 = 𝐷}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wex 1781  wcel 2112  {cab 2779  wrex 3110  Vcvv 3444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-rex 3115  df-v 3446
This theorem is referenced by:  rankcf  10192  sylow1lem2  18720  sylow3lem1  18748  restco  21773
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