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Theorem elintima 38469
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 3354 . . 3 𝑦 ∈ V
21elint2 4619 . 2 (𝑦 {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)} ↔ ∀𝑧 ∈ {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)}𝑦𝑧)
3 elequ2 2159 . . . 4 (𝑧 = 𝑥 → (𝑦𝑧𝑦𝑥))
43ralab2 3523 . . 3 (∀𝑧 ∈ {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)}𝑦𝑧 ↔ ∀𝑥(∃𝑎𝐴 𝑥 = (𝑎𝐵) → 𝑦𝑥))
5 df-rex 3067 . . . . . . 7 (∃𝑎𝐴 𝑥 = (𝑎𝐵) ↔ ∃𝑎(𝑎𝐴𝑥 = (𝑎𝐵)))
65imbi1i 338 . . . . . 6 ((∃𝑎𝐴 𝑥 = (𝑎𝐵) → 𝑦𝑥) ↔ (∃𝑎(𝑎𝐴𝑥 = (𝑎𝐵)) → 𝑦𝑥))
7 19.23v 2023 . . . . . 6 (∀𝑎((𝑎𝐴𝑥 = (𝑎𝐵)) → 𝑦𝑥) ↔ (∃𝑎(𝑎𝐴𝑥 = (𝑎𝐵)) → 𝑦𝑥))
8 simpr 471 . . . . . . . . . 10 ((𝑎𝐴𝑥 = (𝑎𝐵)) → 𝑥 = (𝑎𝐵))
98eleq2d 2836 . . . . . . . . 9 ((𝑎𝐴𝑥 = (𝑎𝐵)) → (𝑦𝑥𝑦 ∈ (𝑎𝐵)))
109pm5.74i 260 . . . . . . . 8 (((𝑎𝐴𝑥 = (𝑎𝐵)) → 𝑦𝑥) ↔ ((𝑎𝐴𝑥 = (𝑎𝐵)) → 𝑦 ∈ (𝑎𝐵)))
111elima 5611 . . . . . . . . . 10 (𝑦 ∈ (𝑎𝐵) ↔ ∃𝑏𝐵 𝑏𝑎𝑦)
12 df-br 4788 . . . . . . . . . . 11 (𝑏𝑎𝑦 ↔ ⟨𝑏, 𝑦⟩ ∈ 𝑎)
1312rexbii 3189 . . . . . . . . . 10 (∃𝑏𝐵 𝑏𝑎𝑦 ↔ ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎)
1411, 13bitri 264 . . . . . . . . 9 (𝑦 ∈ (𝑎𝐵) ↔ ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎)
1514imbi2i 325 . . . . . . . 8 (((𝑎𝐴𝑥 = (𝑎𝐵)) → 𝑦 ∈ (𝑎𝐵)) ↔ ((𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎))
1610, 15bitri 264 . . . . . . 7 (((𝑎𝐴𝑥 = (𝑎𝐵)) → 𝑦𝑥) ↔ ((𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎))
1716albii 1895 . . . . . 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 1895 . . . 4 (∀𝑥(∃𝑎𝐴 𝑥 = (𝑎𝐵) → 𝑦𝑥) ↔ ∀𝑥𝑎((𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎))
20 19.23v 2023 . . . . . . 7 (∀𝑥((𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎) ↔ (∃𝑥(𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎))
21 vex 3354 . . . . . . . . . . 11 𝑎 ∈ V
22 imaexg 7254 . . . . . . . . . . 11 (𝑎 ∈ V → (𝑎𝐵) ∈ V)
2321, 22ax-mp 5 . . . . . . . . . 10 (𝑎𝐵) ∈ V
2423isseti 3361 . . . . . . . . 9 𝑥 𝑥 = (𝑎𝐵)
25 19.42v 2033 . . . . . . . . 9 (∃𝑥(𝑎𝐴𝑥 = (𝑎𝐵)) ↔ (𝑎𝐴 ∧ ∃𝑥 𝑥 = (𝑎𝐵)))
2624, 25mpbiran2 689 . . . . . . . 8 (∃𝑥(𝑎𝐴𝑥 = (𝑎𝐵)) ↔ 𝑎𝐴)
2726imbi1i 338 . . . . . . 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 1895 . . . . 5 (∀𝑎𝑥((𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎) ↔ ∀𝑎(𝑎𝐴 → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎))
30 alcom 2193 . . . . 5 (∀𝑥𝑎((𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎) ↔ ∀𝑎𝑥((𝑎𝐴𝑥 = (𝑎𝐵)) → ∃𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎))
31 df-ral 3066 . . . . 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 382  wal 1629   = wceq 1631  wex 1852  wcel 2145  {cab 2757  wral 3061  wrex 3062  Vcvv 3351  cop 4323   cint 4612   class class class wbr 4787  cima 5253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-int 4613  df-br 4788  df-opab 4848  df-xp 5256  df-cnv 5258  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263
This theorem is referenced by:  intimass  38470  intimag  38472
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