Theorem abrexco 5653

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 2420	. . . . 5 ⊢ (∃𝑦 ∈ {𝑧 ∣ ∃𝑤 ∈ 𝐴 𝑧 = 𝐵}𝑥 = 𝐶 ↔ ∃𝑦(𝑦 ∈ {𝑧 ∣ ∃𝑤 ∈ 𝐴 𝑧 = 𝐵} ∧ 𝑥 = 𝐶))
2		vex 2684	. . . . . . . . 9 ⊢ 𝑦 ∈ V
3		eqeq1 2144	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → (𝑧 = 𝐵 ↔ 𝑦 = 𝐵))
4	3	rexbidv 2436	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (∃𝑤 ∈ 𝐴 𝑧 = 𝐵 ↔ ∃𝑤 ∈ 𝐴 𝑦 = 𝐵))
5	2, 4	elab 2823	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑧 ∣ ∃𝑤 ∈ 𝐴 𝑧 = 𝐵} ↔ ∃𝑤 ∈ 𝐴 𝑦 = 𝐵)
6	5	anbi1i 453	. . . . . . 7 ⊢ ((𝑦 ∈ {𝑧 ∣ ∃𝑤 ∈ 𝐴 𝑧 = 𝐵} ∧ 𝑥 = 𝐶) ↔ (∃𝑤 ∈ 𝐴 𝑦 = 𝐵 ∧ 𝑥 = 𝐶))
7		r19.41v 2585	. . . . . . 7 ⊢ (∃𝑤 ∈ 𝐴 (𝑦 = 𝐵 ∧ 𝑥 = 𝐶) ↔ (∃𝑤 ∈ 𝐴 𝑦 = 𝐵 ∧ 𝑥 = 𝐶))
8	6, 7	bitr4i 186	. . . . . 6 ⊢ ((𝑦 ∈ {𝑧 ∣ ∃𝑤 ∈ 𝐴 𝑧 = 𝐵} ∧ 𝑥 = 𝐶) ↔ ∃𝑤 ∈ 𝐴 (𝑦 = 𝐵 ∧ 𝑥 = 𝐶))
9	8	exbii 1584	. . . . 5 ⊢ (∃𝑦(𝑦 ∈ {𝑧 ∣ ∃𝑤 ∈ 𝐴 𝑧 = 𝐵} ∧ 𝑥 = 𝐶) ↔ ∃𝑦∃𝑤 ∈ 𝐴 (𝑦 = 𝐵 ∧ 𝑥 = 𝐶))
10	1, 9	bitri 183	. . . 4 ⊢ (∃𝑦 ∈ {𝑧 ∣ ∃𝑤 ∈ 𝐴 𝑧 = 𝐵}𝑥 = 𝐶 ↔ ∃𝑦∃𝑤 ∈ 𝐴 (𝑦 = 𝐵 ∧ 𝑥 = 𝐶))
11		rexcom4 2704	. . . 4 ⊢ (∃𝑤 ∈ 𝐴 ∃𝑦(𝑦 = 𝐵 ∧ 𝑥 = 𝐶) ↔ ∃𝑦∃𝑤 ∈ 𝐴 (𝑦 = 𝐵 ∧ 𝑥 = 𝐶))
12	10, 11	bitr4i 186	. . 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 2149	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑥 = 𝐶 ↔ 𝑥 = 𝐷))
16	13, 15	ceqsexv 2720	. . . 4 ⊢ (∃𝑦(𝑦 = 𝐵 ∧ 𝑥 = 𝐶) ↔ 𝑥 = 𝐷)
17	16	rexbii 2440	. . 3 ⊢ (∃𝑤 ∈ 𝐴 ∃𝑦(𝑦 = 𝐵 ∧ 𝑥 = 𝐶) ↔ ∃𝑤 ∈ 𝐴 𝑥 = 𝐷)
18	12, 17	bitri 183	. 2 ⊢ (∃𝑦 ∈ {𝑧 ∣ ∃𝑤 ∈ 𝐴 𝑧 = 𝐵}𝑥 = 𝐶 ↔ ∃𝑤 ∈ 𝐴 𝑥 = 𝐷)
19	18	abbii 2253	1 ⊢ {𝑥 ∣ ∃𝑦 ∈ {𝑧 ∣ ∃𝑤 ∈ 𝐴 𝑧 = 𝐵}𝑥 = 𝐶} = {𝑥 ∣ ∃𝑤 ∈ 𝐴 𝑥 = 𝐷}

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331  ∃wex 1468   ∈ wcel 1480  {cab 2123  ∃wrex 2415  Vcvv 2681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-v 2683

This theorem is referenced by:  restco  12332