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Theorem abrexco 6542
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 2947 . . . . 5 (∃𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵}𝑥 = 𝐶 ↔ ∃𝑦(𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵} ∧ 𝑥 = 𝐶))
2 vex 3234 . . . . . . . . 9 𝑦 ∈ V
3 eqeq1 2655 . . . . . . . . . 10 (𝑧 = 𝑦 → (𝑧 = 𝐵𝑦 = 𝐵))
43rexbidv 3081 . . . . . . . . 9 (𝑧 = 𝑦 → (∃𝑤𝐴 𝑧 = 𝐵 ↔ ∃𝑤𝐴 𝑦 = 𝐵))
52, 4elab 3382 . . . . . . . 8 (𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵} ↔ ∃𝑤𝐴 𝑦 = 𝐵)
65anbi1i 731 . . . . . . 7 ((𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵} ∧ 𝑥 = 𝐶) ↔ (∃𝑤𝐴 𝑦 = 𝐵𝑥 = 𝐶))
7 r19.41v 3118 . . . . . . 7 (∃𝑤𝐴 (𝑦 = 𝐵𝑥 = 𝐶) ↔ (∃𝑤𝐴 𝑦 = 𝐵𝑥 = 𝐶))
86, 7bitr4i 267 . . . . . 6 ((𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵} ∧ 𝑥 = 𝐶) ↔ ∃𝑤𝐴 (𝑦 = 𝐵𝑥 = 𝐶))
98exbii 1814 . . . . 5 (∃𝑦(𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵} ∧ 𝑥 = 𝐶) ↔ ∃𝑦𝑤𝐴 (𝑦 = 𝐵𝑥 = 𝐶))
101, 9bitri 264 . . . 4 (∃𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵}𝑥 = 𝐶 ↔ ∃𝑦𝑤𝐴 (𝑦 = 𝐵𝑥 = 𝐶))
11 rexcom4 3256 . . . 4 (∃𝑤𝐴𝑦(𝑦 = 𝐵𝑥 = 𝐶) ↔ ∃𝑦𝑤𝐴 (𝑦 = 𝐵𝑥 = 𝐶))
1210, 11bitr4i 267 . . 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 2661 . . . . 5 (𝑦 = 𝐵 → (𝑥 = 𝐶𝑥 = 𝐷))
1613, 15ceqsexv 3273 . . . 4 (∃𝑦(𝑦 = 𝐵𝑥 = 𝐶) ↔ 𝑥 = 𝐷)
1716rexbii 3070 . . 3 (∃𝑤𝐴𝑦(𝑦 = 𝐵𝑥 = 𝐶) ↔ ∃𝑤𝐴 𝑥 = 𝐷)
1812, 17bitri 264 . 2 (∃𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵}𝑥 = 𝐶 ↔ ∃𝑤𝐴 𝑥 = 𝐷)
1918abbii 2768 1 {𝑥 ∣ ∃𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵}𝑥 = 𝐶} = {𝑥 ∣ ∃𝑤𝐴 𝑥 = 𝐷}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wex 1744  wcel 2030  {cab 2637  wrex 2942  Vcvv 3231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233
This theorem is referenced by:  rankcf  9637  sylow1lem2  18060  sylow3lem1  18088  restco  21016
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