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Theorem elintima 41999
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 3452 . . 3 𝑦 ∈ V
21elint2 4919 . 2 (𝑦 {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)} ↔ ∀𝑧 ∈ {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)}𝑦𝑧)
3 elequ2 2122 . . . 4 (𝑧 = 𝑥 → (𝑦𝑧𝑦𝑥))
43ralab2 3660 . . 3 (∀𝑧 ∈ {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)}𝑦𝑧 ↔ ∀𝑥(∃𝑎𝐴 𝑥 = (𝑎𝐵) → 𝑦𝑥))
5 df-rex 3075 . . . . . . 7 (∃𝑎𝐴 𝑥 = (𝑎𝐵) ↔ ∃𝑎(𝑎𝐴𝑥 = (𝑎𝐵)))
65imbi1i 350 . . . . . 6 ((∃𝑎𝐴 𝑥 = (𝑎𝐵) → 𝑦𝑥) ↔ (∃𝑎(𝑎𝐴𝑥 = (𝑎𝐵)) → 𝑦𝑥))
7 19.23v 1946 . . . . . 6 (∀𝑎((𝑎𝐴𝑥 = (𝑎𝐵)) → 𝑦𝑥) ↔ (∃𝑎(𝑎𝐴𝑥 = (𝑎𝐵)) → 𝑦𝑥))
8 simpr 486 . . . . . . . . . 10 ((𝑎𝐴𝑥 = (𝑎𝐵)) → 𝑥 = (𝑎𝐵))
98eleq2d 2824 . . . . . . . . 9 ((𝑎𝐴𝑥 = (𝑎𝐵)) → (𝑦𝑥𝑦 ∈ (𝑎𝐵)))
109pm5.74i 271 . . . . . . . 8 (((𝑎𝐴𝑥 = (𝑎𝐵)) → 𝑦𝑥) ↔ ((𝑎𝐴𝑥 = (𝑎𝐵)) → 𝑦 ∈ (𝑎𝐵)))
111elima 6023 . . . . . . . . . 10 (𝑦 ∈ (𝑎𝐵) ↔ ∃𝑏𝐵 𝑏𝑎𝑦)
12 df-br 5111 . . . . . . . . . . 11 (𝑏𝑎𝑦 ↔ ⟨𝑏, 𝑦⟩ ∈ 𝑎)
1312rexbii 3098 . . . . . . . . . 10 (∃𝑏𝐵 𝑏𝑎𝑦 ↔ ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎)
1411, 13bitri 275 . . . . . . . . 9 (𝑦 ∈ (𝑎𝐵) ↔ ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎)
1514imbi2i 336 . . . . . . . 8 (((𝑎𝐴𝑥 = (𝑎𝐵)) → 𝑦 ∈ (𝑎𝐵)) ↔ ((𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎))
1610, 15bitri 275 . . . . . . 7 (((𝑎𝐴𝑥 = (𝑎𝐵)) → 𝑦𝑥) ↔ ((𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎))
1716albii 1822 . . . . . 6 (∀𝑎((𝑎𝐴𝑥 = (𝑎𝐵)) → 𝑦𝑥) ↔ ∀𝑎((𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎))
186, 7, 173bitr2i 299 . . . . 5 ((∃𝑎𝐴 𝑥 = (𝑎𝐵) → 𝑦𝑥) ↔ ∀𝑎((𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎))
1918albii 1822 . . . 4 (∀𝑥(∃𝑎𝐴 𝑥 = (𝑎𝐵) → 𝑦𝑥) ↔ ∀𝑥𝑎((𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎))
20 19.23v 1946 . . . . . . 7 (∀𝑥((𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎) ↔ (∃𝑥(𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎))
21 vex 3452 . . . . . . . . . . 11 𝑎 ∈ V
2221imaex 7858 . . . . . . . . . 10 (𝑎𝐵) ∈ V
2322isseti 3463 . . . . . . . . 9 𝑥 𝑥 = (𝑎𝐵)
24 19.42v 1958 . . . . . . . . 9 (∃𝑥(𝑎𝐴𝑥 = (𝑎𝐵)) ↔ (𝑎𝐴 ∧ ∃𝑥 𝑥 = (𝑎𝐵)))
2523, 24mpbiran2 709 . . . . . . . 8 (∃𝑥(𝑎𝐴𝑥 = (𝑎𝐵)) ↔ 𝑎𝐴)
2625imbi1i 350 . . . . . . 7 ((∃𝑥(𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎) ↔ (𝑎𝐴 → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎))
2720, 26bitri 275 . . . . . 6 (∀𝑥((𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎) ↔ (𝑎𝐴 → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎))
2827albii 1822 . . . . 5 (∀𝑎𝑥((𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎) ↔ ∀𝑎(𝑎𝐴 → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎))
29 alcom 2157 . . . . 5 (∀𝑥𝑎((𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎) ↔ ∀𝑎𝑥((𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎))
30 df-ral 3066 . . . . 5 (∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎 ↔ ∀𝑎(𝑎𝐴 → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎))
3128, 29, 303bitr4i 303 . . . 4 (∀𝑥𝑎((𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎) ↔ ∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎)
3219, 31bitri 275 . . 3 (∀𝑥(∃𝑎𝐴 𝑥 = (𝑎𝐵) → 𝑦𝑥) ↔ ∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎)
334, 32bitri 275 . 2 (∀𝑧 ∈ {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)}𝑦𝑧 ↔ ∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎)
342, 33bitri 275 1 (𝑦 {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)} ↔ ∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wal 1540   = wceq 1542  wex 1782  wcel 2107  {cab 2714  wral 3065  wrex 3074  cop 4597   cint 4912   class class class wbr 5110  cima 5641
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 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
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-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-br 5111  df-opab 5173  df-xp 5644  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651
This theorem is referenced by:  intimass  42000  intimag  42002
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