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Theorem kmlem11 9020
Description: Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 26-Mar-2004.)
Hypothesis
Ref Expression
kmlem9.1 𝐴 = {𝑢 ∣ ∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡}))}
Assertion
Ref Expression
kmlem11 (𝑧𝑥 → (𝑧 𝐴) = (𝑧 (𝑥 ∖ {𝑧})))
Distinct variable groups:   𝑥,𝑧,𝑢,𝑡   𝑧,𝐴
Allowed substitution hints:   𝐴(𝑥,𝑢,𝑡)

Proof of Theorem kmlem11
StepHypRef Expression
1 kmlem9.1 . . . . . 6 𝐴 = {𝑢 ∣ ∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡}))}
21unieqi 4477 . . . . 5 𝐴 = {𝑢 ∣ ∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡}))}
3 vex 3234 . . . . . . 7 𝑡 ∈ V
43difexi 4842 . . . . . 6 (𝑡 (𝑥 ∖ {𝑡})) ∈ V
54dfiun2 4586 . . . . 5 𝑡𝑥 (𝑡 (𝑥 ∖ {𝑡})) = {𝑢 ∣ ∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡}))}
62, 5eqtr4i 2676 . . . 4 𝐴 = 𝑡𝑥 (𝑡 (𝑥 ∖ {𝑡}))
76ineq2i 3844 . . 3 (𝑧 𝐴) = (𝑧 𝑡𝑥 (𝑡 (𝑥 ∖ {𝑡})))
8 iunin2 4616 . . 3 𝑡𝑥 (𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡}))) = (𝑧 𝑡𝑥 (𝑡 (𝑥 ∖ {𝑡})))
97, 8eqtr4i 2676 . 2 (𝑧 𝐴) = 𝑡𝑥 (𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡})))
10 undif2 4077 . . . . . 6 ({𝑧} ∪ (𝑥 ∖ {𝑧})) = ({𝑧} ∪ 𝑥)
11 snssi 4371 . . . . . . 7 (𝑧𝑥 → {𝑧} ⊆ 𝑥)
12 ssequn1 3816 . . . . . . 7 ({𝑧} ⊆ 𝑥 ↔ ({𝑧} ∪ 𝑥) = 𝑥)
1311, 12sylib 208 . . . . . 6 (𝑧𝑥 → ({𝑧} ∪ 𝑥) = 𝑥)
1410, 13syl5req 2698 . . . . 5 (𝑧𝑥𝑥 = ({𝑧} ∪ (𝑥 ∖ {𝑧})))
1514iuneq1d 4577 . . . 4 (𝑧𝑥 𝑡𝑥 (𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡}))) = 𝑡 ∈ ({𝑧} ∪ (𝑥 ∖ {𝑧}))(𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡}))))
16 iunxun 4637 . . . . . 6 𝑡 ∈ ({𝑧} ∪ (𝑥 ∖ {𝑧}))(𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡}))) = ( 𝑡 ∈ {𝑧} (𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡}))) ∪ 𝑡 ∈ (𝑥 ∖ {𝑧})(𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡}))))
17 vex 3234 . . . . . . . 8 𝑧 ∈ V
18 difeq1 3754 . . . . . . . . . 10 (𝑡 = 𝑧 → (𝑡 (𝑥 ∖ {𝑡})) = (𝑧 (𝑥 ∖ {𝑡})))
19 sneq 4220 . . . . . . . . . . . . 13 (𝑡 = 𝑧 → {𝑡} = {𝑧})
2019difeq2d 3761 . . . . . . . . . . . 12 (𝑡 = 𝑧 → (𝑥 ∖ {𝑡}) = (𝑥 ∖ {𝑧}))
2120unieqd 4478 . . . . . . . . . . 11 (𝑡 = 𝑧 (𝑥 ∖ {𝑡}) = (𝑥 ∖ {𝑧}))
2221difeq2d 3761 . . . . . . . . . 10 (𝑡 = 𝑧 → (𝑧 (𝑥 ∖ {𝑡})) = (𝑧 (𝑥 ∖ {𝑧})))
2318, 22eqtrd 2685 . . . . . . . . 9 (𝑡 = 𝑧 → (𝑡 (𝑥 ∖ {𝑡})) = (𝑧 (𝑥 ∖ {𝑧})))
2423ineq2d 3847 . . . . . . . 8 (𝑡 = 𝑧 → (𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡}))) = (𝑧 ∩ (𝑧 (𝑥 ∖ {𝑧}))))
2517, 24iunxsn 4635 . . . . . . 7 𝑡 ∈ {𝑧} (𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡}))) = (𝑧 ∩ (𝑧 (𝑥 ∖ {𝑧})))
2625uneq1i 3796 . . . . . 6 ( 𝑡 ∈ {𝑧} (𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡}))) ∪ 𝑡 ∈ (𝑥 ∖ {𝑧})(𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡})))) = ((𝑧 ∩ (𝑧 (𝑥 ∖ {𝑧}))) ∪ 𝑡 ∈ (𝑥 ∖ {𝑧})(𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡}))))
2716, 26eqtri 2673 . . . . 5 𝑡 ∈ ({𝑧} ∪ (𝑥 ∖ {𝑧}))(𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡}))) = ((𝑧 ∩ (𝑧 (𝑥 ∖ {𝑧}))) ∪ 𝑡 ∈ (𝑥 ∖ {𝑧})(𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡}))))
28 eldifsni 4353 . . . . . . . . . 10 (𝑡 ∈ (𝑥 ∖ {𝑧}) → 𝑡𝑧)
29 incom 3838 . . . . . . . . . . . 12 (𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡}))) = ((𝑡 (𝑥 ∖ {𝑡})) ∩ 𝑧)
30 kmlem4 9013 . . . . . . . . . . . 12 ((𝑧𝑥𝑡𝑧) → ((𝑡 (𝑥 ∖ {𝑡})) ∩ 𝑧) = ∅)
3129, 30syl5eq 2697 . . . . . . . . . . 11 ((𝑧𝑥𝑡𝑧) → (𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡}))) = ∅)
3231ex 449 . . . . . . . . . 10 (𝑧𝑥 → (𝑡𝑧 → (𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡}))) = ∅))
3328, 32syl5 34 . . . . . . . . 9 (𝑧𝑥 → (𝑡 ∈ (𝑥 ∖ {𝑧}) → (𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡}))) = ∅))
3433ralrimiv 2994 . . . . . . . 8 (𝑧𝑥 → ∀𝑡 ∈ (𝑥 ∖ {𝑧})(𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡}))) = ∅)
35 iuneq2 4569 . . . . . . . 8 (∀𝑡 ∈ (𝑥 ∖ {𝑧})(𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡}))) = ∅ → 𝑡 ∈ (𝑥 ∖ {𝑧})(𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡}))) = 𝑡 ∈ (𝑥 ∖ {𝑧})∅)
3634, 35syl 17 . . . . . . 7 (𝑧𝑥 𝑡 ∈ (𝑥 ∖ {𝑧})(𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡}))) = 𝑡 ∈ (𝑥 ∖ {𝑧})∅)
37 iun0 4608 . . . . . . 7 𝑡 ∈ (𝑥 ∖ {𝑧})∅ = ∅
3836, 37syl6eq 2701 . . . . . 6 (𝑧𝑥 𝑡 ∈ (𝑥 ∖ {𝑧})(𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡}))) = ∅)
3938uneq2d 3800 . . . . 5 (𝑧𝑥 → ((𝑧 ∩ (𝑧 (𝑥 ∖ {𝑧}))) ∪ 𝑡 ∈ (𝑥 ∖ {𝑧})(𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡})))) = ((𝑧 ∩ (𝑧 (𝑥 ∖ {𝑧}))) ∪ ∅))
4027, 39syl5eq 2697 . . . 4 (𝑧𝑥 𝑡 ∈ ({𝑧} ∪ (𝑥 ∖ {𝑧}))(𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡}))) = ((𝑧 ∩ (𝑧 (𝑥 ∖ {𝑧}))) ∪ ∅))
4115, 40eqtrd 2685 . . 3 (𝑧𝑥 𝑡𝑥 (𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡}))) = ((𝑧 ∩ (𝑧 (𝑥 ∖ {𝑧}))) ∪ ∅))
42 un0 4000 . . . 4 ((𝑧 ∩ (𝑧 (𝑥 ∖ {𝑧}))) ∪ ∅) = (𝑧 ∩ (𝑧 (𝑥 ∖ {𝑧})))
43 indif 3902 . . . 4 (𝑧 ∩ (𝑧 (𝑥 ∖ {𝑧}))) = (𝑧 (𝑥 ∖ {𝑧}))
4442, 43eqtri 2673 . . 3 ((𝑧 ∩ (𝑧 (𝑥 ∖ {𝑧}))) ∪ ∅) = (𝑧 (𝑥 ∖ {𝑧}))
4541, 44syl6eq 2701 . 2 (𝑧𝑥 𝑡𝑥 (𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡}))) = (𝑧 (𝑥 ∖ {𝑧})))
469, 45syl5eq 2697 1 (𝑧𝑥 → (𝑧 𝐴) = (𝑧 (𝑥 ∖ {𝑧})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  {cab 2637  wne 2823  wral 2941  wrex 2942  cdif 3604  cun 3605  cin 3606  wss 3607  c0 3948  {csn 4210   cuni 4468   ciun 4552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-sn 4211  df-uni 4469  df-iun 4554
This theorem is referenced by:  kmlem12  9021
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