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Theorem kmlem12 8927
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 3699 . . . . . . 7 (𝑡 = 𝑧 → (𝑡 (𝑥 ∖ {𝑡})) = (𝑧 (𝑥 ∖ {𝑡})))
2 sneq 4158 . . . . . . . . . 10 (𝑡 = 𝑧 → {𝑡} = {𝑧})
32difeq2d 3706 . . . . . . . . 9 (𝑡 = 𝑧 → (𝑥 ∖ {𝑡}) = (𝑥 ∖ {𝑧}))
43unieqd 4412 . . . . . . . 8 (𝑡 = 𝑧 (𝑥 ∖ {𝑡}) = (𝑥 ∖ {𝑧}))
54difeq2d 3706 . . . . . . 7 (𝑡 = 𝑧 → (𝑧 (𝑥 ∖ {𝑡})) = (𝑧 (𝑥 ∖ {𝑧})))
61, 5eqtrd 2655 . . . . . 6 (𝑡 = 𝑧 → (𝑡 (𝑥 ∖ {𝑡})) = (𝑧 (𝑥 ∖ {𝑧})))
76neeq1d 2849 . . . . 5 (𝑡 = 𝑧 → ((𝑡 (𝑥 ∖ {𝑡})) ≠ ∅ ↔ (𝑧 (𝑥 ∖ {𝑧})) ≠ ∅))
87cbvralv 3159 . . . 4 (∀𝑡𝑥 (𝑡 (𝑥 ∖ {𝑡})) ≠ ∅ ↔ ∀𝑧𝑥 (𝑧 (𝑥 ∖ {𝑧})) ≠ ∅)
96ineq1d 3791 . . . . . . 7 (𝑡 = 𝑧 → ((𝑡 (𝑥 ∖ {𝑡})) ∩ 𝑦) = ((𝑧 (𝑥 ∖ {𝑧})) ∩ 𝑦))
109eleq2d 2684 . . . . . 6 (𝑡 = 𝑧 → (𝑣 ∈ ((𝑡 (𝑥 ∖ {𝑡})) ∩ 𝑦) ↔ 𝑣 ∈ ((𝑧 (𝑥 ∖ {𝑧})) ∩ 𝑦)))
1110eubidv 2489 . . . . 5 (𝑡 = 𝑧 → (∃!𝑣 𝑣 ∈ ((𝑡 (𝑥 ∖ {𝑡})) ∩ 𝑦) ↔ ∃!𝑣 𝑣 ∈ ((𝑧 (𝑥 ∖ {𝑧})) ∩ 𝑦)))
1211cbvralv 3159 . . . 4 (∀𝑡𝑥 ∃!𝑣 𝑣 ∈ ((𝑡 (𝑥 ∖ {𝑡})) ∩ 𝑦) ↔ ∀𝑧𝑥 ∃!𝑣 𝑣 ∈ ((𝑧 (𝑥 ∖ {𝑧})) ∩ 𝑦))
138, 12imbi12i 340 . . 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 3802 . . . . . . . . . 10 (𝑧 ∩ (𝑦 𝐴)) = (𝑦 ∩ (𝑧 𝐴))
15 incom 3783 . . . . . . . . . 10 (𝑦 ∩ (𝑧 𝐴)) = ((𝑧 𝐴) ∩ 𝑦)
1614, 15eqtri 2643 . . . . . . . . 9 (𝑧 ∩ (𝑦 𝐴)) = ((𝑧 𝐴) ∩ 𝑦)
17 kmlem9.1 . . . . . . . . . . 11 𝐴 = {𝑢 ∣ ∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡}))}
1817kmlem11 8926 . . . . . . . . . 10 (𝑧𝑥 → (𝑧 𝐴) = (𝑧 (𝑥 ∖ {𝑧})))
1918ineq1d 3791 . . . . . . . . 9 (𝑧𝑥 → ((𝑧 𝐴) ∩ 𝑦) = ((𝑧 (𝑥 ∖ {𝑧})) ∩ 𝑦))
2016, 19syl5req 2668 . . . . . . . 8 (𝑧𝑥 → ((𝑧 (𝑥 ∖ {𝑧})) ∩ 𝑦) = (𝑧 ∩ (𝑦 𝐴)))
2120eleq2d 2684 . . . . . . 7 (𝑧𝑥 → (𝑣 ∈ ((𝑧 (𝑥 ∖ {𝑧})) ∩ 𝑦) ↔ 𝑣 ∈ (𝑧 ∩ (𝑦 𝐴))))
2221eubidv 2489 . . . . . 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 243 . . . . 5 (𝑧𝑥 → (∃!𝑣 𝑣 ∈ ((𝑧 (𝑥 ∖ {𝑧})) ∩ 𝑦) → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑦 𝐴)))))
2524ralimia 2945 . . . 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 207 . 2 ((∀𝑡𝑥 (𝑡 (𝑥 ∖ {𝑡})) ≠ ∅ → ∀𝑡𝑥 ∃!𝑣 𝑣 ∈ ((𝑡 (𝑥 ∖ {𝑡})) ∩ 𝑦)) → (∀𝑧𝑥 (𝑧 (𝑥 ∖ {𝑧})) ≠ ∅ → ∀𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑦 𝐴)))))
2817raleqi 3131 . . . 4 (∀𝑧𝐴 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦)) ↔ ∀𝑧 ∈ {𝑢 ∣ ∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡}))} (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦)))
29 df-ral 2912 . . . 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 3189 . . . . . . . . 9 𝑧 ∈ V
31 eqeq1 2625 . . . . . . . . . 10 (𝑢 = 𝑧 → (𝑢 = (𝑡 (𝑥 ∖ {𝑡})) ↔ 𝑧 = (𝑡 (𝑥 ∖ {𝑡}))))
3231rexbidv 3045 . . . . . . . . 9 (𝑢 = 𝑧 → (∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡})) ↔ ∃𝑡𝑥 𝑧 = (𝑡 (𝑥 ∖ {𝑡}))))
3330, 32elab 3333 . . . . . . . 8 (𝑧 ∈ {𝑢 ∣ ∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡}))} ↔ ∃𝑡𝑥 𝑧 = (𝑡 (𝑥 ∖ {𝑡})))
3433imbi1i 339 . . . . . . 7 ((𝑧 ∈ {𝑢 ∣ ∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡}))} → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦))) ↔ (∃𝑡𝑥 𝑧 = (𝑡 (𝑥 ∖ {𝑡})) → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦))))
35 r19.23v 3016 . . . . . . 7 (∀𝑡𝑥 (𝑧 = (𝑡 (𝑥 ∖ {𝑡})) → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦))) ↔ (∃𝑡𝑥 𝑧 = (𝑡 (𝑥 ∖ {𝑡})) → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦))))
3634, 35bitr4i 267 . . . . . 6 ((𝑧 ∈ {𝑢 ∣ ∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡}))} → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦))) ↔ ∀𝑡𝑥 (𝑧 = (𝑡 (𝑥 ∖ {𝑡})) → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦))))
3736albii 1744 . . . . 5 (∀𝑧(𝑧 ∈ {𝑢 ∣ ∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡}))} → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦))) ↔ ∀𝑧𝑡𝑥 (𝑧 = (𝑡 (𝑥 ∖ {𝑡})) → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦))))
38 ralcom4 3210 . . . . 5 (∀𝑡𝑥𝑧(𝑧 = (𝑡 (𝑥 ∖ {𝑡})) → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦))) ↔ ∀𝑧𝑡𝑥 (𝑧 = (𝑡 (𝑥 ∖ {𝑡})) → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦))))
39 vex 3189 . . . . . . . 8 𝑡 ∈ V
4039difexi 4769 . . . . . . 7 (𝑡 (𝑥 ∖ {𝑡})) ∈ V
41 neeq1 2852 . . . . . . . 8 (𝑧 = (𝑡 (𝑥 ∖ {𝑡})) → (𝑧 ≠ ∅ ↔ (𝑡 (𝑥 ∖ {𝑡})) ≠ ∅))
42 ineq1 3785 . . . . . . . . . 10 (𝑧 = (𝑡 (𝑥 ∖ {𝑡})) → (𝑧𝑦) = ((𝑡 (𝑥 ∖ {𝑡})) ∩ 𝑦))
4342eleq2d 2684 . . . . . . . . 9 (𝑧 = (𝑡 (𝑥 ∖ {𝑡})) → (𝑣 ∈ (𝑧𝑦) ↔ 𝑣 ∈ ((𝑡 (𝑥 ∖ {𝑡})) ∩ 𝑦)))
4443eubidv 2489 . . . . . . . 8 (𝑧 = (𝑡 (𝑥 ∖ {𝑡})) → (∃!𝑣 𝑣 ∈ (𝑧𝑦) ↔ ∃!𝑣 𝑣 ∈ ((𝑡 (𝑥 ∖ {𝑡})) ∩ 𝑦)))
4541, 44imbi12d 334 . . . . . . 7 (𝑧 = (𝑡 (𝑥 ∖ {𝑡})) → ((𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦)) ↔ ((𝑡 (𝑥 ∖ {𝑡})) ≠ ∅ → ∃!𝑣 𝑣 ∈ ((𝑡 (𝑥 ∖ {𝑡})) ∩ 𝑦))))
4640, 45ceqsalv 3219 . . . . . 6 (∀𝑧(𝑧 = (𝑡 (𝑥 ∖ {𝑡})) → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦))) ↔ ((𝑡 (𝑥 ∖ {𝑡})) ≠ ∅ → ∃!𝑣 𝑣 ∈ ((𝑡 (𝑥 ∖ {𝑡})) ∩ 𝑦)))
4746ralbii 2974 . . . . 5 (∀𝑡𝑥𝑧(𝑧 = (𝑡 (𝑥 ∖ {𝑡})) → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦))) ↔ ∀𝑡𝑥 ((𝑡 (𝑥 ∖ {𝑡})) ≠ ∅ → ∃!𝑣 𝑣 ∈ ((𝑡 (𝑥 ∖ {𝑡})) ∩ 𝑦)))
4837, 38, 473bitr2i 288 . . . 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 286 . . 3 (∀𝑧𝐴 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦)) ↔ ∀𝑡𝑥 ((𝑡 (𝑥 ∖ {𝑡})) ≠ ∅ → ∃!𝑣 𝑣 ∈ ((𝑡 (𝑥 ∖ {𝑡})) ∩ 𝑦)))
50 ralim 2943 . . 3 (∀𝑡𝑥 ((𝑡 (𝑥 ∖ {𝑡})) ≠ ∅ → ∃!𝑣 𝑣 ∈ ((𝑡 (𝑥 ∖ {𝑡})) ∩ 𝑦)) → (∀𝑡𝑥 (𝑡 (𝑥 ∖ {𝑡})) ≠ ∅ → ∀𝑡𝑥 ∃!𝑣 𝑣 ∈ ((𝑡 (𝑥 ∖ {𝑡})) ∩ 𝑦)))
5149, 50sylbi 207 . 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 1478   = wceq 1480  wcel 1987  ∃!weu 2469  {cab 2607  wne 2790  wral 2907  wrex 2908  cdif 3552  cin 3554  c0 3891  {csn 4148   cuni 4402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-sn 4149  df-uni 4403  df-iun 4487
This theorem is referenced by:  kmlem13  8928
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