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Theorem dfiun2gOLD 4865
Description: Obsolete proof of dfiun2g 4864 as of 11-Aug-2023. (Contributed by NM, 23-Mar-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (Proof modification is discouraged.) (New usage 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 3188 . . . . . 6 𝑥𝑥𝐴 𝐵𝐶
2 rsp 3174 . . . . . . . 8 (∀𝑥𝐴 𝐵𝐶 → (𝑥𝐴𝐵𝐶))
3 clel3g 3595 . . . . . . . 8 (𝐵𝐶 → (𝑧𝐵 ↔ ∃𝑦(𝑦 = 𝐵𝑧𝑦)))
42, 3syl6 35 . . . . . . 7 (∀𝑥𝐴 𝐵𝐶 → (𝑥𝐴 → (𝑧𝐵 ↔ ∃𝑦(𝑦 = 𝐵𝑧𝑦))))
54imp 407 . . . . . 6 ((∀𝑥𝐴 𝐵𝐶𝑥𝐴) → (𝑧𝐵 ↔ ∃𝑦(𝑦 = 𝐵𝑧𝑦)))
61, 5rexbida 3281 . . . . 5 (∀𝑥𝐴 𝐵𝐶 → (∃𝑥𝐴 𝑧𝐵 ↔ ∃𝑥𝐴𝑦(𝑦 = 𝐵𝑧𝑦)))
7 rexcom4 3215 . . . . 5 (∃𝑥𝐴𝑦(𝑦 = 𝐵𝑧𝑦) ↔ ∃𝑦𝑥𝐴 (𝑦 = 𝐵𝑧𝑦))
86, 7syl6bb 288 . . . 4 (∀𝑥𝐴 𝐵𝐶 → (∃𝑥𝐴 𝑧𝐵 ↔ ∃𝑦𝑥𝐴 (𝑦 = 𝐵𝑧𝑦)))
9 r19.41v 3310 . . . . . 6 (∃𝑥𝐴 (𝑦 = 𝐵𝑧𝑦) ↔ (∃𝑥𝐴 𝑦 = 𝐵𝑧𝑦))
109exbii 1833 . . . . 5 (∃𝑦𝑥𝐴 (𝑦 = 𝐵𝑧𝑦) ↔ ∃𝑦(∃𝑥𝐴 𝑦 = 𝐵𝑧𝑦))
11 exancom 1846 . . . . 5 (∃𝑦(∃𝑥𝐴 𝑦 = 𝐵𝑧𝑦) ↔ ∃𝑦(𝑧𝑦 ∧ ∃𝑥𝐴 𝑦 = 𝐵))
1210, 11bitri 276 . . . 4 (∃𝑦𝑥𝐴 (𝑦 = 𝐵𝑧𝑦) ↔ ∃𝑦(𝑧𝑦 ∧ ∃𝑥𝐴 𝑦 = 𝐵))
138, 12syl6bb 288 . . 3 (∀𝑥𝐴 𝐵𝐶 → (∃𝑥𝐴 𝑧𝐵 ↔ ∃𝑦(𝑧𝑦 ∧ ∃𝑥𝐴 𝑦 = 𝐵)))
14 eliun 4835 . . 3 (𝑧 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑧𝐵)
15 eluniab 4762 . . 3 (𝑧 {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ↔ ∃𝑦(𝑧𝑦 ∧ ∃𝑥𝐴 𝑦 = 𝐵))
1613, 14, 153bitr4g 315 . 2 (∀𝑥𝐴 𝐵𝐶 → (𝑧 𝑥𝐴 𝐵𝑧 {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}))
1716eqrdv 2795 1 (∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1525  wex 1765  wcel 2083  {cab 2777  wral 3107  wrex 3108   cuni 4751   ciun 4831
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113  df-v 3442  df-uni 4752  df-iun 4833
This theorem is referenced by: (None)
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