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Theorem elima4 33023
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 3515 . 2 (𝐴 ∈ (𝑅𝐵) → 𝐴 ∈ V)
2 xpeq2 5579 . . . . . . 7 ({𝐴} = ∅ → (𝐵 × {𝐴}) = (𝐵 × ∅))
3 xp0 6018 . . . . . . 7 (𝐵 × ∅) = ∅
42, 3syl6eq 2875 . . . . . 6 ({𝐴} = ∅ → (𝐵 × {𝐴}) = ∅)
54ineq2d 4192 . . . . 5 ({𝐴} = ∅ → (𝑅 ∩ (𝐵 × {𝐴})) = (𝑅 ∩ ∅))
6 in0 4348 . . . . 5 (𝑅 ∩ ∅) = ∅
75, 6syl6eq 2875 . . . 4 ({𝐴} = ∅ → (𝑅 ∩ (𝐵 × {𝐴})) = ∅)
87necon3i 3051 . . 3 ((𝑅 ∩ (𝐵 × {𝐴})) ≠ ∅ → {𝐴} ≠ ∅)
9 snnzb 4657 . . 3 (𝐴 ∈ V ↔ {𝐴} ≠ ∅)
108, 9sylibr 236 . 2 ((𝑅 ∩ (𝐵 × {𝐴})) ≠ ∅ → 𝐴 ∈ V)
11 eleq1 2903 . . 3 (𝑥 = 𝐴 → (𝑥 ∈ (𝑅𝐵) ↔ 𝐴 ∈ (𝑅𝐵)))
12 sneq 4580 . . . . . 6 (𝑥 = 𝐴 → {𝑥} = {𝐴})
1312xpeq2d 5588 . . . . 5 (𝑥 = 𝐴 → (𝐵 × {𝑥}) = (𝐵 × {𝐴}))
1413ineq2d 4192 . . . 4 (𝑥 = 𝐴 → (𝑅 ∩ (𝐵 × {𝑥})) = (𝑅 ∩ (𝐵 × {𝐴})))
1514neeq1d 3078 . . 3 (𝑥 = 𝐴 → ((𝑅 ∩ (𝐵 × {𝑥})) ≠ ∅ ↔ (𝑅 ∩ (𝐵 × {𝐴})) ≠ ∅))
16 elin 4172 . . . . . . 7 (𝑝 ∈ (𝑅 ∩ (𝐵 × {𝑥})) ↔ (𝑝𝑅𝑝 ∈ (𝐵 × {𝑥})))
17 ancom 463 . . . . . . 7 ((𝑝𝑅𝑝 ∈ (𝐵 × {𝑥})) ↔ (𝑝 ∈ (𝐵 × {𝑥}) ∧ 𝑝𝑅))
18 elxp 5581 . . . . . . . 8 (𝑝 ∈ (𝐵 × {𝑥}) ↔ ∃𝑦𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧 ∈ {𝑥})))
1918anbi1i 625 . . . . . . 7 ((𝑝 ∈ (𝐵 × {𝑥}) ∧ 𝑝𝑅) ↔ (∃𝑦𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧 ∈ {𝑥})) ∧ 𝑝𝑅))
2016, 17, 193bitri 299 . . . . . 6 (𝑝 ∈ (𝑅 ∩ (𝐵 × {𝑥})) ↔ (∃𝑦𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧 ∈ {𝑥})) ∧ 𝑝𝑅))
2120exbii 1847 . . . . 5 (∃𝑝 𝑝 ∈ (𝑅 ∩ (𝐵 × {𝑥})) ↔ ∃𝑝(∃𝑦𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧 ∈ {𝑥})) ∧ 𝑝𝑅))
22 anass 471 . . . . . . . . 9 (((𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧 ∈ {𝑥})) ∧ 𝑝𝑅) ↔ (𝑝 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦𝐵𝑧 ∈ {𝑥}) ∧ 𝑝𝑅)))
23222exbii 1848 . . . . . . . 8 (∃𝑦𝑧((𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧 ∈ {𝑥})) ∧ 𝑝𝑅) ↔ ∃𝑦𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦𝐵𝑧 ∈ {𝑥}) ∧ 𝑝𝑅)))
24 19.41vv 1950 . . . . . . . 8 (∃𝑦𝑧((𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧 ∈ {𝑥})) ∧ 𝑝𝑅) ↔ (∃𝑦𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧 ∈ {𝑥})) ∧ 𝑝𝑅))
2523, 24bitr3i 279 . . . . . . 7 (∃𝑦𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦𝐵𝑧 ∈ {𝑥}) ∧ 𝑝𝑅)) ↔ (∃𝑦𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧 ∈ {𝑥})) ∧ 𝑝𝑅))
2625exbii 1847 . . . . . 6 (∃𝑝𝑦𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦𝐵𝑧 ∈ {𝑥}) ∧ 𝑝𝑅)) ↔ ∃𝑝(∃𝑦𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧 ∈ {𝑥})) ∧ 𝑝𝑅))
27 exrot3 2171 . . . . . 6 (∃𝑝𝑦𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦𝐵𝑧 ∈ {𝑥}) ∧ 𝑝𝑅)) ↔ ∃𝑦𝑧𝑝(𝑝 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦𝐵𝑧 ∈ {𝑥}) ∧ 𝑝𝑅)))
2826, 27bitr3i 279 . . . . 5 (∃𝑝(∃𝑦𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧 ∈ {𝑥})) ∧ 𝑝𝑅) ↔ ∃𝑦𝑧𝑝(𝑝 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦𝐵𝑧 ∈ {𝑥}) ∧ 𝑝𝑅)))
29 opex 5359 . . . . . . . . 9 𝑦, 𝑧⟩ ∈ V
30 eleq1 2903 . . . . . . . . . 10 (𝑝 = ⟨𝑦, 𝑧⟩ → (𝑝𝑅 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝑅))
3130anbi2d 630 . . . . . . . . 9 (𝑝 = ⟨𝑦, 𝑧⟩ → (((𝑦𝐵𝑧 ∈ {𝑥}) ∧ 𝑝𝑅) ↔ ((𝑦𝐵𝑧 ∈ {𝑥}) ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑅)))
3229, 31ceqsexv 3544 . . . . . . . 8 (∃𝑝(𝑝 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦𝐵𝑧 ∈ {𝑥}) ∧ 𝑝𝑅)) ↔ ((𝑦𝐵𝑧 ∈ {𝑥}) ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑅))
3332exbii 1847 . . . . . . 7 (∃𝑧𝑝(𝑝 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦𝐵𝑧 ∈ {𝑥}) ∧ 𝑝𝑅)) ↔ ∃𝑧((𝑦𝐵𝑧 ∈ {𝑥}) ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑅))
34 anass 471 . . . . . . . . 9 (((𝑦𝐵𝑧 ∈ {𝑥}) ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑅) ↔ (𝑦𝐵 ∧ (𝑧 ∈ {𝑥} ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑅)))
35 an12 643 . . . . . . . . 9 ((𝑦𝐵 ∧ (𝑧 ∈ {𝑥} ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑅)) ↔ (𝑧 ∈ {𝑥} ∧ (𝑦𝐵 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑅)))
36 velsn 4586 . . . . . . . . . 10 (𝑧 ∈ {𝑥} ↔ 𝑧 = 𝑥)
3736anbi1i 625 . . . . . . . . 9 ((𝑧 ∈ {𝑥} ∧ (𝑦𝐵 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑅)) ↔ (𝑧 = 𝑥 ∧ (𝑦𝐵 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑅)))
3834, 35, 373bitri 299 . . . . . . . 8 (((𝑦𝐵𝑧 ∈ {𝑥}) ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑅) ↔ (𝑧 = 𝑥 ∧ (𝑦𝐵 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑅)))
3938exbii 1847 . . . . . . 7 (∃𝑧((𝑦𝐵𝑧 ∈ {𝑥}) ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑅) ↔ ∃𝑧(𝑧 = 𝑥 ∧ (𝑦𝐵 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑅)))
40 vex 3500 . . . . . . . 8 𝑥 ∈ V
41 opeq2 4807 . . . . . . . . . 10 (𝑧 = 𝑥 → ⟨𝑦, 𝑧⟩ = ⟨𝑦, 𝑥⟩)
4241eleq1d 2900 . . . . . . . . 9 (𝑧 = 𝑥 → (⟨𝑦, 𝑧⟩ ∈ 𝑅 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝑅))
4342anbi2d 630 . . . . . . . 8 (𝑧 = 𝑥 → ((𝑦𝐵 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑅) ↔ (𝑦𝐵 ∧ ⟨𝑦, 𝑥⟩ ∈ 𝑅)))
4440, 43ceqsexv 3544 . . . . . . 7 (∃𝑧(𝑧 = 𝑥 ∧ (𝑦𝐵 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑅)) ↔ (𝑦𝐵 ∧ ⟨𝑦, 𝑥⟩ ∈ 𝑅))
4533, 39, 443bitri 299 . . . . . 6 (∃𝑧𝑝(𝑝 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦𝐵𝑧 ∈ {𝑥}) ∧ 𝑝𝑅)) ↔ (𝑦𝐵 ∧ ⟨𝑦, 𝑥⟩ ∈ 𝑅))
4645exbii 1847 . . . . 5 (∃𝑦𝑧𝑝(𝑝 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦𝐵𝑧 ∈ {𝑥}) ∧ 𝑝𝑅)) ↔ ∃𝑦(𝑦𝐵 ∧ ⟨𝑦, 𝑥⟩ ∈ 𝑅))
4721, 28, 463bitri 299 . . . 4 (∃𝑝 𝑝 ∈ (𝑅 ∩ (𝐵 × {𝑥})) ↔ ∃𝑦(𝑦𝐵 ∧ ⟨𝑦, 𝑥⟩ ∈ 𝑅))
48 n0 4313 . . . 4 ((𝑅 ∩ (𝐵 × {𝑥})) ≠ ∅ ↔ ∃𝑝 𝑝 ∈ (𝑅 ∩ (𝐵 × {𝑥})))
4940elima3 5939 . . . 4 (𝑥 ∈ (𝑅𝐵) ↔ ∃𝑦(𝑦𝐵 ∧ ⟨𝑦, 𝑥⟩ ∈ 𝑅))
5047, 48, 493bitr4ri 306 . . 3 (𝑥 ∈ (𝑅𝐵) ↔ (𝑅 ∩ (𝐵 × {𝑥})) ≠ ∅)
5111, 15, 50vtoclbg 3572 . 2 (𝐴 ∈ V → (𝐴 ∈ (𝑅𝐵) ↔ (𝑅 ∩ (𝐵 × {𝐴})) ≠ ∅))
521, 10, 51pm5.21nii 382 1 (𝐴 ∈ (𝑅𝐵) ↔ (𝑅 ∩ (𝐵 × {𝐴})) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 208  wa 398   = wceq 1536  wex 1779  wcel 2113  wne 3019  Vcvv 3497  cin 3938  c0 4294  {csn 4570  cop 4576   × cxp 5556  cima 5561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-br 5070  df-opab 5132  df-xp 5564  df-rel 5565  df-cnv 5566  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571
This theorem is referenced by: (None)
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