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Theorem bnj1366 34844
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 1147 . . 3 (𝜓𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝜑})
31simp2bi 1146 . . . . 5 (𝜓 → ∀𝑥𝐴 ∃!𝑦𝜑)
4 nfcv 2904 . . . . . . 7 𝑦𝐴
5 nfeu1 2587 . . . . . . 7 𝑦∃!𝑦𝜑
64, 5nfralw 3310 . . . . . 6 𝑦𝑥𝐴 ∃!𝑦𝜑
7 nfra1 3283 . . . . . . . 8 𝑥𝑥𝐴 ∃!𝑦𝜑
8 rspa 3247 . . . . . . . . 9 ((∀𝑥𝐴 ∃!𝑦𝜑𝑥𝐴) → ∃!𝑦𝜑)
9 iota1 6537 . . . . . . . . . 10 (∃!𝑦𝜑 → (𝜑 ↔ (℩𝑦𝜑) = 𝑦))
10 eqcom 2743 . . . . . . . . . 10 ((℩𝑦𝜑) = 𝑦𝑦 = (℩𝑦𝜑))
119, 10bitrdi 287 . . . . . . . . 9 (∃!𝑦𝜑 → (𝜑𝑦 = (℩𝑦𝜑)))
128, 11syl 17 . . . . . . . 8 ((∀𝑥𝐴 ∃!𝑦𝜑𝑥𝐴) → (𝜑𝑦 = (℩𝑦𝜑)))
137, 12rexbida 3271 . . . . . . 7 (∀𝑥𝐴 ∃!𝑦𝜑 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐴 𝑦 = (℩𝑦𝜑)))
14 abid 2717 . . . . . . 7 (𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝜑} ↔ ∃𝑥𝐴 𝜑)
15 eqid 2736 . . . . . . . 8 (𝑥𝐴 ↦ (℩𝑦𝜑)) = (𝑥𝐴 ↦ (℩𝑦𝜑))
16 iotaex 6533 . . . . . . . 8 (℩𝑦𝜑) ∈ V
1715, 16elrnmpti 5972 . . . . . . 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 2212 . . . . 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 6515 . . . . . . 7 𝑦(℩𝑦𝜑)
234, 22nfmpt 5248 . . . . . 6 𝑦(𝑥𝐴 ↦ (℩𝑦𝜑))
2423nfrn 5962 . . . . 5 𝑦ran (𝑥𝐴 ↦ (℩𝑦𝜑))
2521, 24cleqf 2933 . . . 4 ({𝑦 ∣ ∃𝑥𝐴 𝜑} = ran (𝑥𝐴 ↦ (℩𝑦𝜑)) ↔ ∀𝑦(𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝜑} ↔ 𝑦 ∈ ran (𝑥𝐴 ↦ (℩𝑦𝜑))))
2620, 25sylibr 234 . . 3 (𝜓 → {𝑦 ∣ ∃𝑥𝐴 𝜑} = ran (𝑥𝐴 ↦ (℩𝑦𝜑)))
272, 26eqtrd 2776 . 2 (𝜓𝐵 = ran (𝑥𝐴 ↦ (℩𝑦𝜑)))
281simp1bi 1145 . . 3 (𝜓𝐴 ∈ V)
29 mptexg 7242 . . 3 (𝐴 ∈ V → (𝑥𝐴 ↦ (℩𝑦𝜑)) ∈ V)
30 rnexg 7925 . . 3 ((𝑥𝐴 ↦ (℩𝑦𝜑)) ∈ V → ran (𝑥𝐴 ↦ (℩𝑦𝜑)) ∈ V)
3128, 29, 303syl 18 . 2 (𝜓 → ran (𝑥𝐴 ↦ (℩𝑦𝜑)) ∈ V)
3227, 31eqeltrd 2840 1 (𝜓𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086  wal 1537   = wceq 1539  wcel 2107  ∃!weu 2567  {cab 2713  wral 3060  wrex 3069  Vcvv 3479  cmpt 5224  ran crn 5685  cio 6511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568
This theorem is referenced by:  bnj1489  35071
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