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Theorem elintima 37423
Description: Element of intersection of images. (Contributed by RP, 13-Apr-2020.)
Assertion
Ref Expression
elintima (𝑦 {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)} ↔ ∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑦,𝑎   𝐵,𝑏   𝑎,𝑏,𝑥,𝑦
Allowed substitution hints:   𝐴(𝑦,𝑎,𝑏)   𝐵(𝑦,𝑎)

Proof of Theorem elintima
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 vex 3189 . . 3 𝑦 ∈ V
21elint2 4447 . 2 (𝑦 {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)} ↔ ∀𝑧 ∈ {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)}𝑦𝑧)
3 elequ2 2001 . . . 4 (𝑧 = 𝑥 → (𝑦𝑧𝑦𝑥))
43ralab2 3353 . . 3 (∀𝑧 ∈ {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)}𝑦𝑧 ↔ ∀𝑥(∃𝑎𝐴 𝑥 = (𝑎𝐵) → 𝑦𝑥))
5 df-rex 2913 . . . . . . 7 (∃𝑎𝐴 𝑥 = (𝑎𝐵) ↔ ∃𝑎(𝑎𝐴𝑥 = (𝑎𝐵)))
65imbi1i 339 . . . . . 6 ((∃𝑎𝐴 𝑥 = (𝑎𝐵) → 𝑦𝑥) ↔ (∃𝑎(𝑎𝐴𝑥 = (𝑎𝐵)) → 𝑦𝑥))
7 19.23v 1899 . . . . . 6 (∀𝑎((𝑎𝐴𝑥 = (𝑎𝐵)) → 𝑦𝑥) ↔ (∃𝑎(𝑎𝐴𝑥 = (𝑎𝐵)) → 𝑦𝑥))
8 simpr 477 . . . . . . . . . 10 ((𝑎𝐴𝑥 = (𝑎𝐵)) → 𝑥 = (𝑎𝐵))
98eleq2d 2684 . . . . . . . . 9 ((𝑎𝐴𝑥 = (𝑎𝐵)) → (𝑦𝑥𝑦 ∈ (𝑎𝐵)))
109pm5.74i 260 . . . . . . . 8 (((𝑎𝐴𝑥 = (𝑎𝐵)) → 𝑦𝑥) ↔ ((𝑎𝐴𝑥 = (𝑎𝐵)) → 𝑦 ∈ (𝑎𝐵)))
111elima 5430 . . . . . . . . . 10 (𝑦 ∈ (𝑎𝐵) ↔ ∃𝑏𝐵 𝑏𝑎𝑦)
12 df-br 4614 . . . . . . . . . . 11 (𝑏𝑎𝑦 ↔ ⟨𝑏, 𝑦⟩ ∈ 𝑎)
1312rexbii 3034 . . . . . . . . . 10 (∃𝑏𝐵 𝑏𝑎𝑦 ↔ ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎)
1411, 13bitri 264 . . . . . . . . 9 (𝑦 ∈ (𝑎𝐵) ↔ ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎)
1514imbi2i 326 . . . . . . . 8 (((𝑎𝐴𝑥 = (𝑎𝐵)) → 𝑦 ∈ (𝑎𝐵)) ↔ ((𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎))
1610, 15bitri 264 . . . . . . 7 (((𝑎𝐴𝑥 = (𝑎𝐵)) → 𝑦𝑥) ↔ ((𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎))
1716albii 1744 . . . . . 6 (∀𝑎((𝑎𝐴𝑥 = (𝑎𝐵)) → 𝑦𝑥) ↔ ∀𝑎((𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎))
186, 7, 173bitr2i 288 . . . . 5 ((∃𝑎𝐴 𝑥 = (𝑎𝐵) → 𝑦𝑥) ↔ ∀𝑎((𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎))
1918albii 1744 . . . 4 (∀𝑥(∃𝑎𝐴 𝑥 = (𝑎𝐵) → 𝑦𝑥) ↔ ∀𝑥𝑎((𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎))
20 19.23v 1899 . . . . . . 7 (∀𝑥((𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎) ↔ (∃𝑥(𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎))
21 vex 3189 . . . . . . . . . . 11 𝑎 ∈ V
22 imaexg 7050 . . . . . . . . . . 11 (𝑎 ∈ V → (𝑎𝐵) ∈ V)
2321, 22ax-mp 5 . . . . . . . . . 10 (𝑎𝐵) ∈ V
2423isseti 3195 . . . . . . . . 9 𝑥 𝑥 = (𝑎𝐵)
25 19.42v 1915 . . . . . . . . 9 (∃𝑥(𝑎𝐴𝑥 = (𝑎𝐵)) ↔ (𝑎𝐴 ∧ ∃𝑥 𝑥 = (𝑎𝐵)))
2624, 25mpbiran2 953 . . . . . . . 8 (∃𝑥(𝑎𝐴𝑥 = (𝑎𝐵)) ↔ 𝑎𝐴)
2726imbi1i 339 . . . . . . 7 ((∃𝑥(𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎) ↔ (𝑎𝐴 → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎))
2820, 27bitri 264 . . . . . 6 (∀𝑥((𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎) ↔ (𝑎𝐴 → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎))
2928albii 1744 . . . . 5 (∀𝑎𝑥((𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎) ↔ ∀𝑎(𝑎𝐴 → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎))
30 alcom 2034 . . . . 5 (∀𝑥𝑎((𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎) ↔ ∀𝑎𝑥((𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎))
31 df-ral 2912 . . . . 5 (∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎 ↔ ∀𝑎(𝑎𝐴 → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎))
3229, 30, 313bitr4i 292 . . . 4 (∀𝑥𝑎((𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎) ↔ ∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎)
3319, 32bitri 264 . . 3 (∀𝑥(∃𝑎𝐴 𝑥 = (𝑎𝐵) → 𝑦𝑥) ↔ ∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎)
344, 33bitri 264 . 2 (∀𝑧 ∈ {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)}𝑦𝑧 ↔ ∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎)
352, 34bitri 264 1 (𝑦 {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)} ↔ ∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1478   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wral 2907  wrex 2908  Vcvv 3186  cop 4154   cint 4440   class class class wbr 4613  cima 5077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-int 4441  df-br 4614  df-opab 4674  df-xp 5080  df-cnv 5082  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087
This theorem is referenced by:  intimass  37424  intimag  37426
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