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Theorem iinabrex 30335
 Description: Rewriting an indexed intersection into an intersection of its image set. (Contributed by Thierry Arnoux, 15-Jun-2024.)
Assertion
Ref Expression
iinabrex (∀𝑥𝐴 𝐵𝑉 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
Distinct variable groups:   𝑥,𝐴,𝑦   𝑦,𝐵
Allowed substitution hints:   𝐵(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem iinabrex
Dummy variables 𝑡 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfra1 3186 . . . . . . . 8 𝑥𝑥𝐴 𝑡𝐵
2 nfv 1915 . . . . . . . 8 𝑥 𝑡𝑧
3 eleq2 2881 . . . . . . . 8 (𝑧 = 𝐵 → (𝑡𝑧𝑡𝐵))
4 vex 3447 . . . . . . . . 9 𝑧 ∈ V
54a1i 11 . . . . . . . 8 (∀𝑥𝐴 𝑡𝐵𝑧 ∈ V)
6 rspa 3174 . . . . . . . 8 ((∀𝑥𝐴 𝑡𝐵𝑥𝐴) → 𝑡𝐵)
71, 2, 3, 5, 6elabreximd 30281 . . . . . . 7 ((∀𝑥𝐴 𝑡𝐵𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}) → 𝑡𝑧)
87ex 416 . . . . . 6 (∀𝑥𝐴 𝑡𝐵 → (𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧))
98alrimiv 1928 . . . . 5 (∀𝑥𝐴 𝑡𝐵 → ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧))
109adantl 485 . . . 4 ((∀𝑥𝐴 𝐵𝑉 ∧ ∀𝑥𝐴 𝑡𝐵) → ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧))
11 nfra1 3186 . . . . . 6 𝑥𝑥𝐴 𝐵𝑉
122nfci 2942 . . . . . . . . 9 𝑥𝑧
13 nfre1 3268 . . . . . . . . . 10 𝑥𝑥𝐴 𝑦 = 𝐵
1413nfab 2964 . . . . . . . . 9 𝑥{𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
1512, 14nfel 2972 . . . . . . . 8 𝑥 𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
1615, 2nfim 1897 . . . . . . 7 𝑥(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧)
1716nfal 2334 . . . . . 6 𝑥𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧)
1811, 17nfan 1900 . . . . 5 𝑥(∀𝑥𝐴 𝐵𝑉 ∧ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧))
19 rspa 3174 . . . . . . . . 9 ((∀𝑥𝐴 𝐵𝑉𝑥𝐴) → 𝐵𝑉)
2019elexd 3464 . . . . . . . 8 ((∀𝑥𝐴 𝐵𝑉𝑥𝐴) → 𝐵 ∈ V)
2120adantlr 714 . . . . . . 7 (((∀𝑥𝐴 𝐵𝑉 ∧ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧)) ∧ 𝑥𝐴) → 𝐵 ∈ V)
22 simplr 768 . . . . . . 7 (((∀𝑥𝐴 𝐵𝑉 ∧ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧)) ∧ 𝑥𝐴) → ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧))
23 rspe 3266 . . . . . . . . . . . 12 ((𝑥𝐴𝑦 = 𝐵) → ∃𝑥𝐴 𝑦 = 𝐵)
24 tbtru 1546 . . . . . . . . . . . 12 (∃𝑥𝐴 𝑦 = 𝐵 ↔ (∃𝑥𝐴 𝑦 = 𝐵 ↔ ⊤))
2523, 24sylib 221 . . . . . . . . . . 11 ((𝑥𝐴𝑦 = 𝐵) → (∃𝑥𝐴 𝑦 = 𝐵 ↔ ⊤))
2625ex 416 . . . . . . . . . 10 (𝑥𝐴 → (𝑦 = 𝐵 → (∃𝑥𝐴 𝑦 = 𝐵 ↔ ⊤)))
2726alrimiv 1928 . . . . . . . . 9 (𝑥𝐴 → ∀𝑦(𝑦 = 𝐵 → (∃𝑥𝐴 𝑦 = 𝐵 ↔ ⊤)))
2827adantl 485 . . . . . . . 8 (((∀𝑥𝐴 𝐵𝑉 ∧ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧)) ∧ 𝑥𝐴) → ∀𝑦(𝑦 = 𝐵 → (∃𝑥𝐴 𝑦 = 𝐵 ↔ ⊤)))
29 elabgt 3612 . . . . . . . . 9 ((𝐵 ∈ V ∧ ∀𝑦(𝑦 = 𝐵 → (∃𝑥𝐴 𝑦 = 𝐵 ↔ ⊤))) → (𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ↔ ⊤))
30 tbtru 1546 . . . . . . . . 9 (𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ↔ (𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ↔ ⊤))
3129, 30sylibr 237 . . . . . . . 8 ((𝐵 ∈ V ∧ ∀𝑦(𝑦 = 𝐵 → (∃𝑥𝐴 𝑦 = 𝐵 ↔ ⊤))) → 𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
3221, 28, 31syl2anc 587 . . . . . . 7 (((∀𝑥𝐴 𝐵𝑉 ∧ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧)) ∧ 𝑥𝐴) → 𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
33 eleq1 2880 . . . . . . . . . . 11 (𝑧 = 𝐵 → (𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ↔ 𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}))
3433, 3imbi12d 348 . . . . . . . . . 10 (𝑧 = 𝐵 → ((𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧) ↔ (𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝐵)))
3534spcgv 3546 . . . . . . . . 9 (𝐵 ∈ V → (∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧) → (𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝐵)))
3635imp 410 . . . . . . . 8 ((𝐵 ∈ V ∧ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧)) → (𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝐵))
3736imp 410 . . . . . . 7 (((𝐵 ∈ V ∧ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧)) ∧ 𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}) → 𝑡𝐵)
3821, 22, 32, 37syl21anc 836 . . . . . 6 (((∀𝑥𝐴 𝐵𝑉 ∧ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧)) ∧ 𝑥𝐴) → 𝑡𝐵)
3938ex 416 . . . . 5 ((∀𝑥𝐴 𝐵𝑉 ∧ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧)) → (𝑥𝐴𝑡𝐵))
4018, 39ralrimi 3183 . . . 4 ((∀𝑥𝐴 𝐵𝑉 ∧ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧)) → ∀𝑥𝐴 𝑡𝐵)
4110, 40impbida 800 . . 3 (∀𝑥𝐴 𝐵𝑉 → (∀𝑥𝐴 𝑡𝐵 ↔ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧)))
4241abbidv 2865 . 2 (∀𝑥𝐴 𝐵𝑉 → {𝑡 ∣ ∀𝑥𝐴 𝑡𝐵} = {𝑡 ∣ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧)})
43 df-iin 4887 . . 3 𝑥𝐴 𝐵 = {𝑡 ∣ ∀𝑥𝐴 𝑡𝐵}
4443a1i 11 . 2 (∀𝑥𝐴 𝐵𝑉 𝑥𝐴 𝐵 = {𝑡 ∣ ∀𝑥𝐴 𝑡𝐵})
45 df-int 4842 . . 3 {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} = {𝑡 ∣ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧)}
4645a1i 11 . 2 (∀𝑥𝐴 𝐵𝑉 {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} = {𝑡 ∣ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑡𝑧)})
4742, 44, 463eqtr4d 2846 1 (∀𝑥𝐴 𝐵𝑉 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538  ⊤wtru 1539   ∈ wcel 2112  {cab 2779  ∀wral 3109  ∃wrex 3110  Vcvv 3444  ∩ cint 4841  ∩ ciin 4885 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-v 3446  df-int 4842  df-iin 4887 This theorem is referenced by:  intimafv  30473
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