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Theorem dfiun2gOLD 5035
Description: Obsolete version of dfiun2g 5034 as of 11-Dec-2024. (Contributed by NM, 23-Mar-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (Proof shortened by Rohan Ridenour, 11-Aug-2023.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
dfiun2gOLD (∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥,𝑦)

Proof of Theorem dfiun2gOLD
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfra1 3280 . . . . . 6 𝑥𝑥𝐴 𝐵𝐶
2 rspa 3244 . . . . . . 7 ((∀𝑥𝐴 𝐵𝐶𝑥𝐴) → 𝐵𝐶)
3 clel3g 3651 . . . . . . 7 (𝐵𝐶 → (𝑧𝐵 ↔ ∃𝑦(𝑦 = 𝐵𝑧𝑦)))
42, 3syl 17 . . . . . 6 ((∀𝑥𝐴 𝐵𝐶𝑥𝐴) → (𝑧𝐵 ↔ ∃𝑦(𝑦 = 𝐵𝑧𝑦)))
51, 4rexbida 3268 . . . . 5 (∀𝑥𝐴 𝐵𝐶 → (∃𝑥𝐴 𝑧𝐵 ↔ ∃𝑥𝐴𝑦(𝑦 = 𝐵𝑧𝑦)))
6 rexcom4 3284 . . . . 5 (∃𝑥𝐴𝑦(𝑦 = 𝐵𝑧𝑦) ↔ ∃𝑦𝑥𝐴 (𝑦 = 𝐵𝑧𝑦))
75, 6bitrdi 286 . . . 4 (∀𝑥𝐴 𝐵𝐶 → (∃𝑥𝐴 𝑧𝐵 ↔ ∃𝑦𝑥𝐴 (𝑦 = 𝐵𝑧𝑦)))
8 r19.41v 3187 . . . . . 6 (∃𝑥𝐴 (𝑦 = 𝐵𝑧𝑦) ↔ (∃𝑥𝐴 𝑦 = 𝐵𝑧𝑦))
98exbii 1849 . . . . 5 (∃𝑦𝑥𝐴 (𝑦 = 𝐵𝑧𝑦) ↔ ∃𝑦(∃𝑥𝐴 𝑦 = 𝐵𝑧𝑦))
10 exancom 1863 . . . . 5 (∃𝑦(∃𝑥𝐴 𝑦 = 𝐵𝑧𝑦) ↔ ∃𝑦(𝑧𝑦 ∧ ∃𝑥𝐴 𝑦 = 𝐵))
119, 10bitri 274 . . . 4 (∃𝑦𝑥𝐴 (𝑦 = 𝐵𝑧𝑦) ↔ ∃𝑦(𝑧𝑦 ∧ ∃𝑥𝐴 𝑦 = 𝐵))
127, 11bitrdi 286 . . 3 (∀𝑥𝐴 𝐵𝐶 → (∃𝑥𝐴 𝑧𝐵 ↔ ∃𝑦(𝑧𝑦 ∧ ∃𝑥𝐴 𝑦 = 𝐵)))
13 eliun 5002 . . 3 (𝑧 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑧𝐵)
14 eluniab 4924 . . 3 (𝑧 {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ↔ ∃𝑦(𝑧𝑦 ∧ ∃𝑥𝐴 𝑦 = 𝐵))
1512, 13, 143bitr4g 313 . 2 (∀𝑥𝐴 𝐵𝐶 → (𝑧 𝑥𝐴 𝐵𝑧 {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}))
1615eqrdv 2729 1 (∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1540  wex 1780  wcel 2105  {cab 2708  wral 3060  wrex 3069   cuni 4909   ciun 4998
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-v 3475  df-uni 4910  df-iun 5000
This theorem is referenced by: (None)
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