Theorem abrexco 7005

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 = 𝐵 → 𝐶 = 𝐷)

Assertion

Ref	Expression
abrexco	⊢ {𝑥 ∣ ∃𝑦 ∈ {𝑧 ∣ ∃𝑤 ∈ 𝐴 𝑧 = 𝐵}𝑥 = 𝐶} = {𝑥 ∣ ∃𝑤 ∈ 𝐴 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

Step	Hyp	Ref	Expression
1		df-rex 3146	. . . . 5 ⊢ (∃𝑦 ∈ {𝑧 ∣ ∃𝑤 ∈ 𝐴 𝑧 = 𝐵}𝑥 = 𝐶 ↔ ∃𝑦(𝑦 ∈ {𝑧 ∣ ∃𝑤 ∈ 𝐴 𝑧 = 𝐵} ∧ 𝑥 = 𝐶))
2		vex 3499	. . . . . . . . 9 ⊢ 𝑦 ∈ V
3		eqeq1 2827	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → (𝑧 = 𝐵 ↔ 𝑦 = 𝐵))
4	3	rexbidv 3299	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (∃𝑤 ∈ 𝐴 𝑧 = 𝐵 ↔ ∃𝑤 ∈ 𝐴 𝑦 = 𝐵))
5	2, 4	elab 3669	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑧 ∣ ∃𝑤 ∈ 𝐴 𝑧 = 𝐵} ↔ ∃𝑤 ∈ 𝐴 𝑦 = 𝐵)
6	5	anbi1i 625	. . . . . . 7 ⊢ ((𝑦 ∈ {𝑧 ∣ ∃𝑤 ∈ 𝐴 𝑧 = 𝐵} ∧ 𝑥 = 𝐶) ↔ (∃𝑤 ∈ 𝐴 𝑦 = 𝐵 ∧ 𝑥 = 𝐶))
7		r19.41v 3349	. . . . . . 7 ⊢ (∃𝑤 ∈ 𝐴 (𝑦 = 𝐵 ∧ 𝑥 = 𝐶) ↔ (∃𝑤 ∈ 𝐴 𝑦 = 𝐵 ∧ 𝑥 = 𝐶))
8	6, 7	bitr4i 280	. . . . . 6 ⊢ ((𝑦 ∈ {𝑧 ∣ ∃𝑤 ∈ 𝐴 𝑧 = 𝐵} ∧ 𝑥 = 𝐶) ↔ ∃𝑤 ∈ 𝐴 (𝑦 = 𝐵 ∧ 𝑥 = 𝐶))
9	8	exbii 1848	. . . . 5 ⊢ (∃𝑦(𝑦 ∈ {𝑧 ∣ ∃𝑤 ∈ 𝐴 𝑧 = 𝐵} ∧ 𝑥 = 𝐶) ↔ ∃𝑦∃𝑤 ∈ 𝐴 (𝑦 = 𝐵 ∧ 𝑥 = 𝐶))
10	1, 9	bitri 277	. . . 4 ⊢ (∃𝑦 ∈ {𝑧 ∣ ∃𝑤 ∈ 𝐴 𝑧 = 𝐵}𝑥 = 𝐶 ↔ ∃𝑦∃𝑤 ∈ 𝐴 (𝑦 = 𝐵 ∧ 𝑥 = 𝐶))
11		rexcom4 3251	. . . 4 ⊢ (∃𝑤 ∈ 𝐴 ∃𝑦(𝑦 = 𝐵 ∧ 𝑥 = 𝐶) ↔ ∃𝑦∃𝑤 ∈ 𝐴 (𝑦 = 𝐵 ∧ 𝑥 = 𝐶))
12	10, 11	bitr4i 280	. . 3 ⊢ (∃𝑦 ∈ {𝑧 ∣ ∃𝑤 ∈ 𝐴 𝑧 = 𝐵}𝑥 = 𝐶 ↔ ∃𝑤 ∈ 𝐴 ∃𝑦(𝑦 = 𝐵 ∧ 𝑥 = 𝐶))
13		abrexco.1	. . . . 5 ⊢ 𝐵 ∈ V
14		abrexco.2	. . . . . 6 ⊢ (𝑦 = 𝐵 → 𝐶 = 𝐷)
15	14	eqeq2d 2834	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑥 = 𝐶 ↔ 𝑥 = 𝐷))
16	13, 15	ceqsexv 3543	. . . 4 ⊢ (∃𝑦(𝑦 = 𝐵 ∧ 𝑥 = 𝐶) ↔ 𝑥 = 𝐷)
17	16	rexbii 3249	. . 3 ⊢ (∃𝑤 ∈ 𝐴 ∃𝑦(𝑦 = 𝐵 ∧ 𝑥 = 𝐶) ↔ ∃𝑤 ∈ 𝐴 𝑥 = 𝐷)
18	12, 17	bitri 277	. 2 ⊢ (∃𝑦 ∈ {𝑧 ∣ ∃𝑤 ∈ 𝐴 𝑧 = 𝐵}𝑥 = 𝐶 ↔ ∃𝑤 ∈ 𝐴 𝑥 = 𝐷)
19	18	abbii 2888	1 ⊢ {𝑥 ∣ ∃𝑦 ∈ {𝑧 ∣ ∃𝑤 ∈ 𝐴 𝑧 = 𝐵}𝑥 = 𝐶} = {𝑥 ∣ ∃𝑤 ∈ 𝐴 𝑥 = 𝐷}

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114  {cab 2801  ∃wrex 3141  Vcvv 3496

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-rex 3146  df-v 3498

This theorem is referenced by:  rankcf  10201  sylow1lem2  18726  sylow3lem1  18754  restco  21774