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Theorem bnj1366 33840
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj1366.1 (𝜓 ↔ (𝐴 ∈ V ∧ ∀𝑥𝐴 ∃!𝑦𝜑𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝜑}))
Assertion
Ref Expression
bnj1366 (𝜓𝐵 ∈ V)
Distinct variable group:   𝑥,𝐴,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem bnj1366
StepHypRef Expression
1 bnj1366.1 . . . 4 (𝜓 ↔ (𝐴 ∈ V ∧ ∀𝑥𝐴 ∃!𝑦𝜑𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝜑}))
21simp3bi 1148 . . 3 (𝜓𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝜑})
31simp2bi 1147 . . . . 5 (𝜓 → ∀𝑥𝐴 ∃!𝑦𝜑)
4 nfcv 2904 . . . . . . 7 𝑦𝐴
5 nfeu1 2583 . . . . . . 7 𝑦∃!𝑦𝜑
64, 5nfralw 3309 . . . . . 6 𝑦𝑥𝐴 ∃!𝑦𝜑
7 nfra1 3282 . . . . . . . 8 𝑥𝑥𝐴 ∃!𝑦𝜑
8 rspa 3246 . . . . . . . . 9 ((∀𝑥𝐴 ∃!𝑦𝜑𝑥𝐴) → ∃!𝑦𝜑)
9 iota1 6521 . . . . . . . . . 10 (∃!𝑦𝜑 → (𝜑 ↔ (℩𝑦𝜑) = 𝑦))
10 eqcom 2740 . . . . . . . . . 10 ((℩𝑦𝜑) = 𝑦𝑦 = (℩𝑦𝜑))
119, 10bitrdi 287 . . . . . . . . 9 (∃!𝑦𝜑 → (𝜑𝑦 = (℩𝑦𝜑)))
128, 11syl 17 . . . . . . . 8 ((∀𝑥𝐴 ∃!𝑦𝜑𝑥𝐴) → (𝜑𝑦 = (℩𝑦𝜑)))
137, 12rexbida 3270 . . . . . . 7 (∀𝑥𝐴 ∃!𝑦𝜑 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐴 𝑦 = (℩𝑦𝜑)))
14 abid 2714 . . . . . . 7 (𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝜑} ↔ ∃𝑥𝐴 𝜑)
15 eqid 2733 . . . . . . . 8 (𝑥𝐴 ↦ (℩𝑦𝜑)) = (𝑥𝐴 ↦ (℩𝑦𝜑))
16 iotaex 6517 . . . . . . . 8 (℩𝑦𝜑) ∈ V
1715, 16elrnmpti 5960 . . . . . . 7 (𝑦 ∈ ran (𝑥𝐴 ↦ (℩𝑦𝜑)) ↔ ∃𝑥𝐴 𝑦 = (℩𝑦𝜑))
1813, 14, 173bitr4g 314 . . . . . 6 (∀𝑥𝐴 ∃!𝑦𝜑 → (𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝜑} ↔ 𝑦 ∈ ran (𝑥𝐴 ↦ (℩𝑦𝜑))))
196, 18alrimi 2207 . . . . 5 (∀𝑥𝐴 ∃!𝑦𝜑 → ∀𝑦(𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝜑} ↔ 𝑦 ∈ ran (𝑥𝐴 ↦ (℩𝑦𝜑))))
203, 19syl 17 . . . 4 (𝜓 → ∀𝑦(𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝜑} ↔ 𝑦 ∈ ran (𝑥𝐴 ↦ (℩𝑦𝜑))))
21 nfab1 2906 . . . . 5 𝑦{𝑦 ∣ ∃𝑥𝐴 𝜑}
22 nfiota1 6498 . . . . . . 7 𝑦(℩𝑦𝜑)
234, 22nfmpt 5256 . . . . . 6 𝑦(𝑥𝐴 ↦ (℩𝑦𝜑))
2423nfrn 5952 . . . . 5 𝑦ran (𝑥𝐴 ↦ (℩𝑦𝜑))
2521, 24cleqf 2935 . . . 4 ({𝑦 ∣ ∃𝑥𝐴 𝜑} = ran (𝑥𝐴 ↦ (℩𝑦𝜑)) ↔ ∀𝑦(𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝜑} ↔ 𝑦 ∈ ran (𝑥𝐴 ↦ (℩𝑦𝜑))))
2620, 25sylibr 233 . . 3 (𝜓 → {𝑦 ∣ ∃𝑥𝐴 𝜑} = ran (𝑥𝐴 ↦ (℩𝑦𝜑)))
272, 26eqtrd 2773 . 2 (𝜓𝐵 = ran (𝑥𝐴 ↦ (℩𝑦𝜑)))
281simp1bi 1146 . . 3 (𝜓𝐴 ∈ V)
29 mptexg 7223 . . 3 (𝐴 ∈ V → (𝑥𝐴 ↦ (℩𝑦𝜑)) ∈ V)
30 rnexg 7895 . . 3 ((𝑥𝐴 ↦ (℩𝑦𝜑)) ∈ V → ran (𝑥𝐴 ↦ (℩𝑦𝜑)) ∈ V)
3128, 29, 303syl 18 . 2 (𝜓 → ran (𝑥𝐴 ↦ (℩𝑦𝜑)) ∈ V)
3227, 31eqeltrd 2834 1 (𝜓𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088  wal 1540   = wceq 1542  wcel 2107  ∃!weu 2563  {cab 2710  wral 3062  wrex 3071  Vcvv 3475  cmpt 5232  ran crn 5678  cio 6494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552
This theorem is referenced by:  bnj1489  34067
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