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Theorem abrexco 6994
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 3141 . . . . 5 (∃𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵}𝑥 = 𝐶 ↔ ∃𝑦(𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵} ∧ 𝑥 = 𝐶))
2 vex 3495 . . . . . . . . 9 𝑦 ∈ V
3 eqeq1 2822 . . . . . . . . . 10 (𝑧 = 𝑦 → (𝑧 = 𝐵𝑦 = 𝐵))
43rexbidv 3294 . . . . . . . . 9 (𝑧 = 𝑦 → (∃𝑤𝐴 𝑧 = 𝐵 ↔ ∃𝑤𝐴 𝑦 = 𝐵))
52, 4elab 3664 . . . . . . . 8 (𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵} ↔ ∃𝑤𝐴 𝑦 = 𝐵)
65anbi1i 623 . . . . . . 7 ((𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵} ∧ 𝑥 = 𝐶) ↔ (∃𝑤𝐴 𝑦 = 𝐵𝑥 = 𝐶))
7 r19.41v 3344 . . . . . . 7 (∃𝑤𝐴 (𝑦 = 𝐵𝑥 = 𝐶) ↔ (∃𝑤𝐴 𝑦 = 𝐵𝑥 = 𝐶))
86, 7bitr4i 279 . . . . . 6 ((𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵} ∧ 𝑥 = 𝐶) ↔ ∃𝑤𝐴 (𝑦 = 𝐵𝑥 = 𝐶))
98exbii 1839 . . . . 5 (∃𝑦(𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵} ∧ 𝑥 = 𝐶) ↔ ∃𝑦𝑤𝐴 (𝑦 = 𝐵𝑥 = 𝐶))
101, 9bitri 276 . . . 4 (∃𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵}𝑥 = 𝐶 ↔ ∃𝑦𝑤𝐴 (𝑦 = 𝐵𝑥 = 𝐶))
11 rexcom4 3246 . . . 4 (∃𝑤𝐴𝑦(𝑦 = 𝐵𝑥 = 𝐶) ↔ ∃𝑦𝑤𝐴 (𝑦 = 𝐵𝑥 = 𝐶))
1210, 11bitr4i 279 . . 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 2829 . . . . 5 (𝑦 = 𝐵 → (𝑥 = 𝐶𝑥 = 𝐷))
1613, 15ceqsexv 3539 . . . 4 (∃𝑦(𝑦 = 𝐵𝑥 = 𝐶) ↔ 𝑥 = 𝐷)
1716rexbii 3244 . . 3 (∃𝑤𝐴𝑦(𝑦 = 𝐵𝑥 = 𝐶) ↔ ∃𝑤𝐴 𝑥 = 𝐷)
1812, 17bitri 276 . 2 (∃𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵}𝑥 = 𝐶 ↔ ∃𝑤𝐴 𝑥 = 𝐷)
1918abbii 2883 1 {𝑥 ∣ ∃𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵}𝑥 = 𝐶} = {𝑥 ∣ ∃𝑤𝐴 𝑥 = 𝐷}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wex 1771  wcel 2105  {cab 2796  wrex 3136  Vcvv 3492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-v 3494
This theorem is referenced by:  rankcf  10187  sylow1lem2  18653  sylow3lem1  18681  restco  21700
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