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Theorem elintima 43225
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 3465 . . 3 𝑦 ∈ V
21elint2 4957 . 2 (𝑦 {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)} ↔ ∀𝑧 ∈ {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)}𝑦𝑧)
3 elequ2 2113 . . . 4 (𝑧 = 𝑥 → (𝑦𝑧𝑦𝑥))
43ralab2 3689 . . 3 (∀𝑧 ∈ {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)}𝑦𝑧 ↔ ∀𝑥(∃𝑎𝐴 𝑥 = (𝑎𝐵) → 𝑦𝑥))
5 df-rex 3060 . . . . . . 7 (∃𝑎𝐴 𝑥 = (𝑎𝐵) ↔ ∃𝑎(𝑎𝐴𝑥 = (𝑎𝐵)))
65imbi1i 348 . . . . . 6 ((∃𝑎𝐴 𝑥 = (𝑎𝐵) → 𝑦𝑥) ↔ (∃𝑎(𝑎𝐴𝑥 = (𝑎𝐵)) → 𝑦𝑥))
7 19.23v 1937 . . . . . 6 (∀𝑎((𝑎𝐴𝑥 = (𝑎𝐵)) → 𝑦𝑥) ↔ (∃𝑎(𝑎𝐴𝑥 = (𝑎𝐵)) → 𝑦𝑥))
8 simpr 483 . . . . . . . . . 10 ((𝑎𝐴𝑥 = (𝑎𝐵)) → 𝑥 = (𝑎𝐵))
98eleq2d 2811 . . . . . . . . 9 ((𝑎𝐴𝑥 = (𝑎𝐵)) → (𝑦𝑥𝑦 ∈ (𝑎𝐵)))
109pm5.74i 270 . . . . . . . 8 (((𝑎𝐴𝑥 = (𝑎𝐵)) → 𝑦𝑥) ↔ ((𝑎𝐴𝑥 = (𝑎𝐵)) → 𝑦 ∈ (𝑎𝐵)))
111elima 6069 . . . . . . . . . 10 (𝑦 ∈ (𝑎𝐵) ↔ ∃𝑏𝐵 𝑏𝑎𝑦)
12 df-br 5150 . . . . . . . . . . 11 (𝑏𝑎𝑦 ↔ ⟨𝑏, 𝑦⟩ ∈ 𝑎)
1312rexbii 3083 . . . . . . . . . 10 (∃𝑏𝐵 𝑏𝑎𝑦 ↔ ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎)
1411, 13bitri 274 . . . . . . . . 9 (𝑦 ∈ (𝑎𝐵) ↔ ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎)
1514imbi2i 335 . . . . . . . 8 (((𝑎𝐴𝑥 = (𝑎𝐵)) → 𝑦 ∈ (𝑎𝐵)) ↔ ((𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎))
1610, 15bitri 274 . . . . . . 7 (((𝑎𝐴𝑥 = (𝑎𝐵)) → 𝑦𝑥) ↔ ((𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎))
1716albii 1813 . . . . . 6 (∀𝑎((𝑎𝐴𝑥 = (𝑎𝐵)) → 𝑦𝑥) ↔ ∀𝑎((𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎))
186, 7, 173bitr2i 298 . . . . 5 ((∃𝑎𝐴 𝑥 = (𝑎𝐵) → 𝑦𝑥) ↔ ∀𝑎((𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎))
1918albii 1813 . . . 4 (∀𝑥(∃𝑎𝐴 𝑥 = (𝑎𝐵) → 𝑦𝑥) ↔ ∀𝑥𝑎((𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎))
20 19.23v 1937 . . . . . . 7 (∀𝑥((𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎) ↔ (∃𝑥(𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎))
21 vex 3465 . . . . . . . . . . 11 𝑎 ∈ V
2221imaex 7922 . . . . . . . . . 10 (𝑎𝐵) ∈ V
2322isseti 3477 . . . . . . . . 9 𝑥 𝑥 = (𝑎𝐵)
24 19.42v 1949 . . . . . . . . 9 (∃𝑥(𝑎𝐴𝑥 = (𝑎𝐵)) ↔ (𝑎𝐴 ∧ ∃𝑥 𝑥 = (𝑎𝐵)))
2523, 24mpbiran2 708 . . . . . . . 8 (∃𝑥(𝑎𝐴𝑥 = (𝑎𝐵)) ↔ 𝑎𝐴)
2625imbi1i 348 . . . . . . 7 ((∃𝑥(𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎) ↔ (𝑎𝐴 → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎))
2720, 26bitri 274 . . . . . 6 (∀𝑥((𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎) ↔ (𝑎𝐴 → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎))
2827albii 1813 . . . . 5 (∀𝑎𝑥((𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎) ↔ ∀𝑎(𝑎𝐴 → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎))
29 alcom 2148 . . . . 5 (∀𝑥𝑎((𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎) ↔ ∀𝑎𝑥((𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎))
30 df-ral 3051 . . . . 5 (∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎 ↔ ∀𝑎(𝑎𝐴 → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎))
3128, 29, 303bitr4i 302 . . . 4 (∀𝑥𝑎((𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎) ↔ ∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎)
3219, 31bitri 274 . . 3 (∀𝑥(∃𝑎𝐴 𝑥 = (𝑎𝐵) → 𝑦𝑥) ↔ ∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎)
334, 32bitri 274 . 2 (∀𝑧 ∈ {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)}𝑦𝑧 ↔ ∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎)
342, 33bitri 274 1 (𝑦 {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)} ↔ ∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wal 1531   = wceq 1533  wex 1773  wcel 2098  {cab 2702  wral 3050  wrex 3059  cop 4636   cint 4950   class class class wbr 5149  cima 5681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-br 5150  df-opab 5212  df-xp 5684  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691
This theorem is referenced by:  intimass  43226  intimag  43228
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