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Theorem kmlem12 10156
Description: Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 27-Mar-2004.)
Hypothesis
Ref Expression
kmlem9.1 𝐴 = {𝑢 ∣ ∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡}))}
Assertion
Ref Expression
kmlem12 (∀𝑧𝑥 (𝑧 (𝑥 ∖ {𝑧})) ≠ ∅ → (∀𝑧𝐴 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦)) → ∀𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑦 𝐴)))))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑣,𝑢,𝑡   𝑦,𝐴,𝑧,𝑣
Allowed substitution hints:   𝐴(𝑥,𝑢,𝑡)

Proof of Theorem kmlem12
StepHypRef Expression
1 difeq1 4116 . . . . . . 7 (𝑡 = 𝑧 → (𝑡 (𝑥 ∖ {𝑡})) = (𝑧 (𝑥 ∖ {𝑡})))
2 sneq 4639 . . . . . . . . . 10 (𝑡 = 𝑧 → {𝑡} = {𝑧})
32difeq2d 4123 . . . . . . . . 9 (𝑡 = 𝑧 → (𝑥 ∖ {𝑡}) = (𝑥 ∖ {𝑧}))
43unieqd 4923 . . . . . . . 8 (𝑡 = 𝑧 (𝑥 ∖ {𝑡}) = (𝑥 ∖ {𝑧}))
54difeq2d 4123 . . . . . . 7 (𝑡 = 𝑧 → (𝑧 (𝑥 ∖ {𝑡})) = (𝑧 (𝑥 ∖ {𝑧})))
61, 5eqtrd 2773 . . . . . 6 (𝑡 = 𝑧 → (𝑡 (𝑥 ∖ {𝑡})) = (𝑧 (𝑥 ∖ {𝑧})))
76neeq1d 3001 . . . . 5 (𝑡 = 𝑧 → ((𝑡 (𝑥 ∖ {𝑡})) ≠ ∅ ↔ (𝑧 (𝑥 ∖ {𝑧})) ≠ ∅))
87cbvralvw 3235 . . . 4 (∀𝑡𝑥 (𝑡 (𝑥 ∖ {𝑡})) ≠ ∅ ↔ ∀𝑧𝑥 (𝑧 (𝑥 ∖ {𝑧})) ≠ ∅)
96ineq1d 4212 . . . . . . 7 (𝑡 = 𝑧 → ((𝑡 (𝑥 ∖ {𝑡})) ∩ 𝑦) = ((𝑧 (𝑥 ∖ {𝑧})) ∩ 𝑦))
109eleq2d 2820 . . . . . 6 (𝑡 = 𝑧 → (𝑣 ∈ ((𝑡 (𝑥 ∖ {𝑡})) ∩ 𝑦) ↔ 𝑣 ∈ ((𝑧 (𝑥 ∖ {𝑧})) ∩ 𝑦)))
1110eubidv 2581 . . . . 5 (𝑡 = 𝑧 → (∃!𝑣 𝑣 ∈ ((𝑡 (𝑥 ∖ {𝑡})) ∩ 𝑦) ↔ ∃!𝑣 𝑣 ∈ ((𝑧 (𝑥 ∖ {𝑧})) ∩ 𝑦)))
1211cbvralvw 3235 . . . 4 (∀𝑡𝑥 ∃!𝑣 𝑣 ∈ ((𝑡 (𝑥 ∖ {𝑡})) ∩ 𝑦) ↔ ∀𝑧𝑥 ∃!𝑣 𝑣 ∈ ((𝑧 (𝑥 ∖ {𝑧})) ∩ 𝑦))
138, 12imbi12i 351 . . 3 ((∀𝑡𝑥 (𝑡 (𝑥 ∖ {𝑡})) ≠ ∅ → ∀𝑡𝑥 ∃!𝑣 𝑣 ∈ ((𝑡 (𝑥 ∖ {𝑡})) ∩ 𝑦)) ↔ (∀𝑧𝑥 (𝑧 (𝑥 ∖ {𝑧})) ≠ ∅ → ∀𝑧𝑥 ∃!𝑣 𝑣 ∈ ((𝑧 (𝑥 ∖ {𝑧})) ∩ 𝑦)))
14 in12 4221 . . . . . . . . . 10 (𝑧 ∩ (𝑦 𝐴)) = (𝑦 ∩ (𝑧 𝐴))
15 incom 4202 . . . . . . . . . 10 (𝑦 ∩ (𝑧 𝐴)) = ((𝑧 𝐴) ∩ 𝑦)
1614, 15eqtri 2761 . . . . . . . . 9 (𝑧 ∩ (𝑦 𝐴)) = ((𝑧 𝐴) ∩ 𝑦)
17 kmlem9.1 . . . . . . . . . . 11 𝐴 = {𝑢 ∣ ∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡}))}
1817kmlem11 10155 . . . . . . . . . 10 (𝑧𝑥 → (𝑧 𝐴) = (𝑧 (𝑥 ∖ {𝑧})))
1918ineq1d 4212 . . . . . . . . 9 (𝑧𝑥 → ((𝑧 𝐴) ∩ 𝑦) = ((𝑧 (𝑥 ∖ {𝑧})) ∩ 𝑦))
2016, 19eqtr2id 2786 . . . . . . . 8 (𝑧𝑥 → ((𝑧 (𝑥 ∖ {𝑧})) ∩ 𝑦) = (𝑧 ∩ (𝑦 𝐴)))
2120eleq2d 2820 . . . . . . 7 (𝑧𝑥 → (𝑣 ∈ ((𝑧 (𝑥 ∖ {𝑧})) ∩ 𝑦) ↔ 𝑣 ∈ (𝑧 ∩ (𝑦 𝐴))))
2221eubidv 2581 . . . . . 6 (𝑧𝑥 → (∃!𝑣 𝑣 ∈ ((𝑧 (𝑥 ∖ {𝑧})) ∩ 𝑦) ↔ ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑦 𝐴))))
23 ax-1 6 . . . . . 6 (∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑦 𝐴)) → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑦 𝐴))))
2422, 23syl6bi 253 . . . . 5 (𝑧𝑥 → (∃!𝑣 𝑣 ∈ ((𝑧 (𝑥 ∖ {𝑧})) ∩ 𝑦) → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑦 𝐴)))))
2524ralimia 3081 . . . 4 (∀𝑧𝑥 ∃!𝑣 𝑣 ∈ ((𝑧 (𝑥 ∖ {𝑧})) ∩ 𝑦) → ∀𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑦 𝐴))))
2625imim2i 16 . . 3 ((∀𝑧𝑥 (𝑧 (𝑥 ∖ {𝑧})) ≠ ∅ → ∀𝑧𝑥 ∃!𝑣 𝑣 ∈ ((𝑧 (𝑥 ∖ {𝑧})) ∩ 𝑦)) → (∀𝑧𝑥 (𝑧 (𝑥 ∖ {𝑧})) ≠ ∅ → ∀𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑦 𝐴)))))
2713, 26sylbi 216 . 2 ((∀𝑡𝑥 (𝑡 (𝑥 ∖ {𝑡})) ≠ ∅ → ∀𝑡𝑥 ∃!𝑣 𝑣 ∈ ((𝑡 (𝑥 ∖ {𝑡})) ∩ 𝑦)) → (∀𝑧𝑥 (𝑧 (𝑥 ∖ {𝑧})) ≠ ∅ → ∀𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑦 𝐴)))))
2817raleqi 3324 . . . 4 (∀𝑧𝐴 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦)) ↔ ∀𝑧 ∈ {𝑢 ∣ ∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡}))} (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦)))
29 df-ral 3063 . . . 4 (∀𝑧 ∈ {𝑢 ∣ ∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡}))} (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦)) ↔ ∀𝑧(𝑧 ∈ {𝑢 ∣ ∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡}))} → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦))))
30 vex 3479 . . . . . . . . 9 𝑧 ∈ V
31 eqeq1 2737 . . . . . . . . . 10 (𝑢 = 𝑧 → (𝑢 = (𝑡 (𝑥 ∖ {𝑡})) ↔ 𝑧 = (𝑡 (𝑥 ∖ {𝑡}))))
3231rexbidv 3179 . . . . . . . . 9 (𝑢 = 𝑧 → (∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡})) ↔ ∃𝑡𝑥 𝑧 = (𝑡 (𝑥 ∖ {𝑡}))))
3330, 32elab 3669 . . . . . . . 8 (𝑧 ∈ {𝑢 ∣ ∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡}))} ↔ ∃𝑡𝑥 𝑧 = (𝑡 (𝑥 ∖ {𝑡})))
3433imbi1i 350 . . . . . . 7 ((𝑧 ∈ {𝑢 ∣ ∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡}))} → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦))) ↔ (∃𝑡𝑥 𝑧 = (𝑡 (𝑥 ∖ {𝑡})) → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦))))
35 r19.23v 3183 . . . . . . 7 (∀𝑡𝑥 (𝑧 = (𝑡 (𝑥 ∖ {𝑡})) → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦))) ↔ (∃𝑡𝑥 𝑧 = (𝑡 (𝑥 ∖ {𝑡})) → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦))))
3634, 35bitr4i 278 . . . . . 6 ((𝑧 ∈ {𝑢 ∣ ∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡}))} → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦))) ↔ ∀𝑡𝑥 (𝑧 = (𝑡 (𝑥 ∖ {𝑡})) → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦))))
3736albii 1822 . . . . 5 (∀𝑧(𝑧 ∈ {𝑢 ∣ ∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡}))} → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦))) ↔ ∀𝑧𝑡𝑥 (𝑧 = (𝑡 (𝑥 ∖ {𝑡})) → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦))))
38 ralcom4 3284 . . . . 5 (∀𝑡𝑥𝑧(𝑧 = (𝑡 (𝑥 ∖ {𝑡})) → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦))) ↔ ∀𝑧𝑡𝑥 (𝑧 = (𝑡 (𝑥 ∖ {𝑡})) → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦))))
39 vex 3479 . . . . . . . 8 𝑡 ∈ V
4039difexi 5329 . . . . . . 7 (𝑡 (𝑥 ∖ {𝑡})) ∈ V
41 neeq1 3004 . . . . . . . 8 (𝑧 = (𝑡 (𝑥 ∖ {𝑡})) → (𝑧 ≠ ∅ ↔ (𝑡 (𝑥 ∖ {𝑡})) ≠ ∅))
42 ineq1 4206 . . . . . . . . . 10 (𝑧 = (𝑡 (𝑥 ∖ {𝑡})) → (𝑧𝑦) = ((𝑡 (𝑥 ∖ {𝑡})) ∩ 𝑦))
4342eleq2d 2820 . . . . . . . . 9 (𝑧 = (𝑡 (𝑥 ∖ {𝑡})) → (𝑣 ∈ (𝑧𝑦) ↔ 𝑣 ∈ ((𝑡 (𝑥 ∖ {𝑡})) ∩ 𝑦)))
4443eubidv 2581 . . . . . . . 8 (𝑧 = (𝑡 (𝑥 ∖ {𝑡})) → (∃!𝑣 𝑣 ∈ (𝑧𝑦) ↔ ∃!𝑣 𝑣 ∈ ((𝑡 (𝑥 ∖ {𝑡})) ∩ 𝑦)))
4541, 44imbi12d 345 . . . . . . 7 (𝑧 = (𝑡 (𝑥 ∖ {𝑡})) → ((𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦)) ↔ ((𝑡 (𝑥 ∖ {𝑡})) ≠ ∅ → ∃!𝑣 𝑣 ∈ ((𝑡 (𝑥 ∖ {𝑡})) ∩ 𝑦))))
4640, 45ceqsalv 3512 . . . . . 6 (∀𝑧(𝑧 = (𝑡 (𝑥 ∖ {𝑡})) → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦))) ↔ ((𝑡 (𝑥 ∖ {𝑡})) ≠ ∅ → ∃!𝑣 𝑣 ∈ ((𝑡 (𝑥 ∖ {𝑡})) ∩ 𝑦)))
4746ralbii 3094 . . . . 5 (∀𝑡𝑥𝑧(𝑧 = (𝑡 (𝑥 ∖ {𝑡})) → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦))) ↔ ∀𝑡𝑥 ((𝑡 (𝑥 ∖ {𝑡})) ≠ ∅ → ∃!𝑣 𝑣 ∈ ((𝑡 (𝑥 ∖ {𝑡})) ∩ 𝑦)))
4837, 38, 473bitr2i 299 . . . 4 (∀𝑧(𝑧 ∈ {𝑢 ∣ ∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡}))} → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦))) ↔ ∀𝑡𝑥 ((𝑡 (𝑥 ∖ {𝑡})) ≠ ∅ → ∃!𝑣 𝑣 ∈ ((𝑡 (𝑥 ∖ {𝑡})) ∩ 𝑦)))
4928, 29, 483bitri 297 . . 3 (∀𝑧𝐴 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦)) ↔ ∀𝑡𝑥 ((𝑡 (𝑥 ∖ {𝑡})) ≠ ∅ → ∃!𝑣 𝑣 ∈ ((𝑡 (𝑥 ∖ {𝑡})) ∩ 𝑦)))
50 ralim 3087 . . 3 (∀𝑡𝑥 ((𝑡 (𝑥 ∖ {𝑡})) ≠ ∅ → ∃!𝑣 𝑣 ∈ ((𝑡 (𝑥 ∖ {𝑡})) ∩ 𝑦)) → (∀𝑡𝑥 (𝑡 (𝑥 ∖ {𝑡})) ≠ ∅ → ∀𝑡𝑥 ∃!𝑣 𝑣 ∈ ((𝑡 (𝑥 ∖ {𝑡})) ∩ 𝑦)))
5149, 50sylbi 216 . 2 (∀𝑧𝐴 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦)) → (∀𝑡𝑥 (𝑡 (𝑥 ∖ {𝑡})) ≠ ∅ → ∀𝑡𝑥 ∃!𝑣 𝑣 ∈ ((𝑡 (𝑥 ∖ {𝑡})) ∩ 𝑦)))
5227, 51syl11 33 1 (∀𝑧𝑥 (𝑧 (𝑥 ∖ {𝑧})) ≠ ∅ → (∀𝑧𝐴 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦)) → ∀𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑦 𝐴)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1540   = wceq 1542  wcel 2107  ∃!weu 2563  {cab 2710  wne 2941  wral 3062  wrex 3071  cdif 3946  cin 3948  c0 4323  {csn 4629   cuni 4909
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-11 2155  ax-ext 2704  ax-sep 5300
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-sn 4630  df-uni 4910  df-iun 5000
This theorem is referenced by:  kmlem13  10157
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