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Theorem elima4 35280
Description: Quantifier-free expression saying that a class is a member of an image. (Contributed by Scott Fenton, 8-May-2018.)
Assertion
Ref Expression
elima4 (𝐴 ∈ (𝑅𝐵) ↔ (𝑅 ∩ (𝐵 × {𝐴})) ≠ ∅)

Proof of Theorem elima4
Dummy variables 𝑥 𝑝 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3487 . 2 (𝐴 ∈ (𝑅𝐵) → 𝐴 ∈ V)
2 xpeq2 5690 . . . . . . 7 ({𝐴} = ∅ → (𝐵 × {𝐴}) = (𝐵 × ∅))
3 xp0 6150 . . . . . . 7 (𝐵 × ∅) = ∅
42, 3eqtrdi 2782 . . . . . 6 ({𝐴} = ∅ → (𝐵 × {𝐴}) = ∅)
54ineq2d 4207 . . . . 5 ({𝐴} = ∅ → (𝑅 ∩ (𝐵 × {𝐴})) = (𝑅 ∩ ∅))
6 in0 4386 . . . . 5 (𝑅 ∩ ∅) = ∅
75, 6eqtrdi 2782 . . . 4 ({𝐴} = ∅ → (𝑅 ∩ (𝐵 × {𝐴})) = ∅)
87necon3i 2967 . . 3 ((𝑅 ∩ (𝐵 × {𝐴})) ≠ ∅ → {𝐴} ≠ ∅)
9 snnzb 4717 . . 3 (𝐴 ∈ V ↔ {𝐴} ≠ ∅)
108, 9sylibr 233 . 2 ((𝑅 ∩ (𝐵 × {𝐴})) ≠ ∅ → 𝐴 ∈ V)
11 eleq1 2815 . . 3 (𝑥 = 𝐴 → (𝑥 ∈ (𝑅𝐵) ↔ 𝐴 ∈ (𝑅𝐵)))
12 sneq 4633 . . . . . 6 (𝑥 = 𝐴 → {𝑥} = {𝐴})
1312xpeq2d 5699 . . . . 5 (𝑥 = 𝐴 → (𝐵 × {𝑥}) = (𝐵 × {𝐴}))
1413ineq2d 4207 . . . 4 (𝑥 = 𝐴 → (𝑅 ∩ (𝐵 × {𝑥})) = (𝑅 ∩ (𝐵 × {𝐴})))
1514neeq1d 2994 . . 3 (𝑥 = 𝐴 → ((𝑅 ∩ (𝐵 × {𝑥})) ≠ ∅ ↔ (𝑅 ∩ (𝐵 × {𝐴})) ≠ ∅))
16 elin 3959 . . . . . . 7 (𝑝 ∈ (𝑅 ∩ (𝐵 × {𝑥})) ↔ (𝑝𝑅𝑝 ∈ (𝐵 × {𝑥})))
17 ancom 460 . . . . . . 7 ((𝑝𝑅𝑝 ∈ (𝐵 × {𝑥})) ↔ (𝑝 ∈ (𝐵 × {𝑥}) ∧ 𝑝𝑅))
18 elxp 5692 . . . . . . . 8 (𝑝 ∈ (𝐵 × {𝑥}) ↔ ∃𝑦𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧 ∈ {𝑥})))
1918anbi1i 623 . . . . . . 7 ((𝑝 ∈ (𝐵 × {𝑥}) ∧ 𝑝𝑅) ↔ (∃𝑦𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧 ∈ {𝑥})) ∧ 𝑝𝑅))
2016, 17, 193bitri 297 . . . . . 6 (𝑝 ∈ (𝑅 ∩ (𝐵 × {𝑥})) ↔ (∃𝑦𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧 ∈ {𝑥})) ∧ 𝑝𝑅))
2120exbii 1842 . . . . 5 (∃𝑝 𝑝 ∈ (𝑅 ∩ (𝐵 × {𝑥})) ↔ ∃𝑝(∃𝑦𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧 ∈ {𝑥})) ∧ 𝑝𝑅))
22 anass 468 . . . . . . . . 9 (((𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧 ∈ {𝑥})) ∧ 𝑝𝑅) ↔ (𝑝 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦𝐵𝑧 ∈ {𝑥}) ∧ 𝑝𝑅)))
23222exbii 1843 . . . . . . . 8 (∃𝑦𝑧((𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧 ∈ {𝑥})) ∧ 𝑝𝑅) ↔ ∃𝑦𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦𝐵𝑧 ∈ {𝑥}) ∧ 𝑝𝑅)))
24 19.41vv 1946 . . . . . . . 8 (∃𝑦𝑧((𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧 ∈ {𝑥})) ∧ 𝑝𝑅) ↔ (∃𝑦𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧 ∈ {𝑥})) ∧ 𝑝𝑅))
2523, 24bitr3i 277 . . . . . . 7 (∃𝑦𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦𝐵𝑧 ∈ {𝑥}) ∧ 𝑝𝑅)) ↔ (∃𝑦𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧 ∈ {𝑥})) ∧ 𝑝𝑅))
2625exbii 1842 . . . . . 6 (∃𝑝𝑦𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦𝐵𝑧 ∈ {𝑥}) ∧ 𝑝𝑅)) ↔ ∃𝑝(∃𝑦𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧 ∈ {𝑥})) ∧ 𝑝𝑅))
27 exrot3 2157 . . . . . 6 (∃𝑝𝑦𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦𝐵𝑧 ∈ {𝑥}) ∧ 𝑝𝑅)) ↔ ∃𝑦𝑧𝑝(𝑝 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦𝐵𝑧 ∈ {𝑥}) ∧ 𝑝𝑅)))
2826, 27bitr3i 277 . . . . 5 (∃𝑝(∃𝑦𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧 ∈ {𝑥})) ∧ 𝑝𝑅) ↔ ∃𝑦𝑧𝑝(𝑝 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦𝐵𝑧 ∈ {𝑥}) ∧ 𝑝𝑅)))
29 opex 5457 . . . . . . . . 9 𝑦, 𝑧⟩ ∈ V
30 eleq1 2815 . . . . . . . . . 10 (𝑝 = ⟨𝑦, 𝑧⟩ → (𝑝𝑅 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝑅))
3130anbi2d 628 . . . . . . . . 9 (𝑝 = ⟨𝑦, 𝑧⟩ → (((𝑦𝐵𝑧 ∈ {𝑥}) ∧ 𝑝𝑅) ↔ ((𝑦𝐵𝑧 ∈ {𝑥}) ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑅)))
3229, 31ceqsexv 3520 . . . . . . . 8 (∃𝑝(𝑝 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦𝐵𝑧 ∈ {𝑥}) ∧ 𝑝𝑅)) ↔ ((𝑦𝐵𝑧 ∈ {𝑥}) ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑅))
3332exbii 1842 . . . . . . 7 (∃𝑧𝑝(𝑝 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦𝐵𝑧 ∈ {𝑥}) ∧ 𝑝𝑅)) ↔ ∃𝑧((𝑦𝐵𝑧 ∈ {𝑥}) ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑅))
34 anass 468 . . . . . . . . 9 (((𝑦𝐵𝑧 ∈ {𝑥}) ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑅) ↔ (𝑦𝐵 ∧ (𝑧 ∈ {𝑥} ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑅)))
35 an12 642 . . . . . . . . 9 ((𝑦𝐵 ∧ (𝑧 ∈ {𝑥} ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑅)) ↔ (𝑧 ∈ {𝑥} ∧ (𝑦𝐵 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑅)))
36 velsn 4639 . . . . . . . . . 10 (𝑧 ∈ {𝑥} ↔ 𝑧 = 𝑥)
3736anbi1i 623 . . . . . . . . 9 ((𝑧 ∈ {𝑥} ∧ (𝑦𝐵 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑅)) ↔ (𝑧 = 𝑥 ∧ (𝑦𝐵 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑅)))
3834, 35, 373bitri 297 . . . . . . . 8 (((𝑦𝐵𝑧 ∈ {𝑥}) ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑅) ↔ (𝑧 = 𝑥 ∧ (𝑦𝐵 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑅)))
3938exbii 1842 . . . . . . 7 (∃𝑧((𝑦𝐵𝑧 ∈ {𝑥}) ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑅) ↔ ∃𝑧(𝑧 = 𝑥 ∧ (𝑦𝐵 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑅)))
40 vex 3472 . . . . . . . 8 𝑥 ∈ V
41 opeq2 4869 . . . . . . . . . 10 (𝑧 = 𝑥 → ⟨𝑦, 𝑧⟩ = ⟨𝑦, 𝑥⟩)
4241eleq1d 2812 . . . . . . . . 9 (𝑧 = 𝑥 → (⟨𝑦, 𝑧⟩ ∈ 𝑅 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝑅))
4342anbi2d 628 . . . . . . . 8 (𝑧 = 𝑥 → ((𝑦𝐵 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑅) ↔ (𝑦𝐵 ∧ ⟨𝑦, 𝑥⟩ ∈ 𝑅)))
4440, 43ceqsexv 3520 . . . . . . 7 (∃𝑧(𝑧 = 𝑥 ∧ (𝑦𝐵 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑅)) ↔ (𝑦𝐵 ∧ ⟨𝑦, 𝑥⟩ ∈ 𝑅))
4533, 39, 443bitri 297 . . . . . 6 (∃𝑧𝑝(𝑝 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦𝐵𝑧 ∈ {𝑥}) ∧ 𝑝𝑅)) ↔ (𝑦𝐵 ∧ ⟨𝑦, 𝑥⟩ ∈ 𝑅))
4645exbii 1842 . . . . 5 (∃𝑦𝑧𝑝(𝑝 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦𝐵𝑧 ∈ {𝑥}) ∧ 𝑝𝑅)) ↔ ∃𝑦(𝑦𝐵 ∧ ⟨𝑦, 𝑥⟩ ∈ 𝑅))
4721, 28, 463bitri 297 . . . 4 (∃𝑝 𝑝 ∈ (𝑅 ∩ (𝐵 × {𝑥})) ↔ ∃𝑦(𝑦𝐵 ∧ ⟨𝑦, 𝑥⟩ ∈ 𝑅))
48 n0 4341 . . . 4 ((𝑅 ∩ (𝐵 × {𝑥})) ≠ ∅ ↔ ∃𝑝 𝑝 ∈ (𝑅 ∩ (𝐵 × {𝑥})))
4940elima3 6059 . . . 4 (𝑥 ∈ (𝑅𝐵) ↔ ∃𝑦(𝑦𝐵 ∧ ⟨𝑦, 𝑥⟩ ∈ 𝑅))
5047, 48, 493bitr4ri 304 . . 3 (𝑥 ∈ (𝑅𝐵) ↔ (𝑅 ∩ (𝐵 × {𝑥})) ≠ ∅)
5111, 15, 50vtoclbg 3539 . 2 (𝐴 ∈ V → (𝐴 ∈ (𝑅𝐵) ↔ (𝑅 ∩ (𝐵 × {𝐴})) ≠ ∅))
521, 10, 51pm5.21nii 378 1 (𝐴 ∈ (𝑅𝐵) ↔ (𝑅 ∩ (𝐵 × {𝐴})) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1533  wex 1773  wcel 2098  wne 2934  Vcvv 3468  cin 3942  c0 4317  {csn 4623  cop 4629   × cxp 5667  cima 5672
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682
This theorem is referenced by: (None)
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