MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  abrexex2g Structured version   Visualization version   GIF version

Theorem abrexex2g 7310
Description: Existence of an existentially restricted class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
abrexex2g ((𝐴𝑉 ∧ ∀𝑥𝐴 {𝑦𝜑} ∈ 𝑊) → {𝑦 ∣ ∃𝑥𝐴 𝜑} ∈ V)
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝑉,𝑦   𝑥,𝑊,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem abrexex2g
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfv 1992 . . . 4 𝑧𝑥𝐴 𝜑
2 nfcv 2902 . . . . 5 𝑦𝐴
3 nfs1v 2574 . . . . 5 𝑦[𝑧 / 𝑦]𝜑
42, 3nfrex 3145 . . . 4 𝑦𝑥𝐴 [𝑧 / 𝑦]𝜑
5 sbequ12 2258 . . . . 5 (𝑦 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑦]𝜑))
65rexbidv 3190 . . . 4 (𝑦 = 𝑧 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐴 [𝑧 / 𝑦]𝜑))
71, 4, 6cbvab 2884 . . 3 {𝑦 ∣ ∃𝑥𝐴 𝜑} = {𝑧 ∣ ∃𝑥𝐴 [𝑧 / 𝑦]𝜑}
8 df-clab 2747 . . . . 5 (𝑧 ∈ {𝑦𝜑} ↔ [𝑧 / 𝑦]𝜑)
98rexbii 3179 . . . 4 (∃𝑥𝐴 𝑧 ∈ {𝑦𝜑} ↔ ∃𝑥𝐴 [𝑧 / 𝑦]𝜑)
109abbii 2877 . . 3 {𝑧 ∣ ∃𝑥𝐴 𝑧 ∈ {𝑦𝜑}} = {𝑧 ∣ ∃𝑥𝐴 [𝑧 / 𝑦]𝜑}
117, 10eqtr4i 2785 . 2 {𝑦 ∣ ∃𝑥𝐴 𝜑} = {𝑧 ∣ ∃𝑥𝐴 𝑧 ∈ {𝑦𝜑}}
12 df-iun 4674 . . 3 𝑥𝐴 {𝑦𝜑} = {𝑧 ∣ ∃𝑥𝐴 𝑧 ∈ {𝑦𝜑}}
13 iunexg 7309 . . 3 ((𝐴𝑉 ∧ ∀𝑥𝐴 {𝑦𝜑} ∈ 𝑊) → 𝑥𝐴 {𝑦𝜑} ∈ V)
1412, 13syl5eqelr 2844 . 2 ((𝐴𝑉 ∧ ∀𝑥𝐴 {𝑦𝜑} ∈ 𝑊) → {𝑧 ∣ ∃𝑥𝐴 𝑧 ∈ {𝑦𝜑}} ∈ V)
1511, 14syl5eqel 2843 1 ((𝐴𝑉 ∧ ∀𝑥𝐴 {𝑦𝜑} ∈ 𝑊) → {𝑦 ∣ ∃𝑥𝐴 𝜑} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  [wsb 2046  wcel 2139  {cab 2746  wral 3050  wrex 3051  Vcvv 3340   ciun 4672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057
This theorem is referenced by:  abrexex2  7314  ptrescn  21664  sdclem2  33869  sdclem1  33870  sprval  42257
  Copyright terms: Public domain W3C validator