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Theorem abrexex2g 7654
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 1906 . . . 4 𝑧𝑥𝐴 𝜑
2 nfcv 2974 . . . . 5 𝑦𝐴
3 nfs1v 2264 . . . . 5 𝑦[𝑧 / 𝑦]𝜑
42, 3nfrex 3306 . . . 4 𝑦𝑥𝐴 [𝑧 / 𝑦]𝜑
5 sbequ12 2243 . . . . 5 (𝑦 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑦]𝜑))
65rexbidv 3294 . . . 4 (𝑦 = 𝑧 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐴 [𝑧 / 𝑦]𝜑))
71, 4, 6cbvabw 2887 . . 3 {𝑦 ∣ ∃𝑥𝐴 𝜑} = {𝑧 ∣ ∃𝑥𝐴 [𝑧 / 𝑦]𝜑}
8 df-clab 2797 . . . . 5 (𝑧 ∈ {𝑦𝜑} ↔ [𝑧 / 𝑦]𝜑)
98rexbii 3244 . . . 4 (∃𝑥𝐴 𝑧 ∈ {𝑦𝜑} ↔ ∃𝑥𝐴 [𝑧 / 𝑦]𝜑)
109abbii 2883 . . 3 {𝑧 ∣ ∃𝑥𝐴 𝑧 ∈ {𝑦𝜑}} = {𝑧 ∣ ∃𝑥𝐴 [𝑧 / 𝑦]𝜑}
117, 10eqtr4i 2844 . 2 {𝑦 ∣ ∃𝑥𝐴 𝜑} = {𝑧 ∣ ∃𝑥𝐴 𝑧 ∈ {𝑦𝜑}}
12 df-iun 4912 . . 3 𝑥𝐴 {𝑦𝜑} = {𝑧 ∣ ∃𝑥𝐴 𝑧 ∈ {𝑦𝜑}}
13 iunexg 7653 . . 3 ((𝐴𝑉 ∧ ∀𝑥𝐴 {𝑦𝜑} ∈ 𝑊) → 𝑥𝐴 {𝑦𝜑} ∈ V)
1412, 13eqeltrrid 2915 . 2 ((𝐴𝑉 ∧ ∀𝑥𝐴 {𝑦𝜑} ∈ 𝑊) → {𝑧 ∣ ∃𝑥𝐴 𝑧 ∈ {𝑦𝜑}} ∈ V)
1511, 14eqeltrid 2914 1 ((𝐴𝑉 ∧ ∀𝑥𝐴 {𝑦𝜑} ∈ 𝑊) → {𝑦 ∣ ∃𝑥𝐴 𝜑} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  [wsb 2060  wcel 2105  {cab 2796  wral 3135  wrex 3136  Vcvv 3492   ciun 4910
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  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356
This theorem is referenced by:  abrexex2  7659  ptrescn  22175  satfvsuclem1  32503  satf0suclem  32519  fmlasuc0  32528  sdclem2  34898  sdclem1  34899  sprval  43518  prprval  43553
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