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Theorem kmlem11 10157
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 4921 . . . . 5 𝐴 = {𝑢 ∣ ∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡}))}
3 vex 3478 . . . . . . 7 𝑡 ∈ V
43difexi 5328 . . . . . 6 (𝑡 (𝑥 ∖ {𝑡})) ∈ V
54dfiun2 5036 . . . . 5 𝑡𝑥 (𝑡 (𝑥 ∖ {𝑡})) = {𝑢 ∣ ∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡}))}
62, 5eqtr4i 2763 . . . 4 𝐴 = 𝑡𝑥 (𝑡 (𝑥 ∖ {𝑡}))
76ineq2i 4209 . . 3 (𝑧 𝐴) = (𝑧 𝑡𝑥 (𝑡 (𝑥 ∖ {𝑡})))
8 iunin2 5074 . . 3 𝑡𝑥 (𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡}))) = (𝑧 𝑡𝑥 (𝑡 (𝑥 ∖ {𝑡})))
97, 8eqtr4i 2763 . 2 (𝑧 𝐴) = 𝑡𝑥 (𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡})))
10 undif2 4476 . . . . . 6 ({𝑧} ∪ (𝑥 ∖ {𝑧})) = ({𝑧} ∪ 𝑥)
11 snssi 4811 . . . . . . 7 (𝑧𝑥 → {𝑧} ⊆ 𝑥)
12 ssequn1 4180 . . . . . . 7 ({𝑧} ⊆ 𝑥 ↔ ({𝑧} ∪ 𝑥) = 𝑥)
1311, 12sylib 217 . . . . . 6 (𝑧𝑥 → ({𝑧} ∪ 𝑥) = 𝑥)
1410, 13eqtr2id 2785 . . . . 5 (𝑧𝑥𝑥 = ({𝑧} ∪ (𝑥 ∖ {𝑧})))
1514iuneq1d 5024 . . . 4 (𝑧𝑥 𝑡𝑥 (𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡}))) = 𝑡 ∈ ({𝑧} ∪ (𝑥 ∖ {𝑧}))(𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡}))))
16 iunxun 5097 . . . . . 6 𝑡 ∈ ({𝑧} ∪ (𝑥 ∖ {𝑧}))(𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡}))) = ( 𝑡 ∈ {𝑧} (𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡}))) ∪ 𝑡 ∈ (𝑥 ∖ {𝑧})(𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡}))))
17 vex 3478 . . . . . . . 8 𝑧 ∈ V
18 difeq1 4115 . . . . . . . . . 10 (𝑡 = 𝑧 → (𝑡 (𝑥 ∖ {𝑡})) = (𝑧 (𝑥 ∖ {𝑡})))
19 sneq 4638 . . . . . . . . . . . . 13 (𝑡 = 𝑧 → {𝑡} = {𝑧})
2019difeq2d 4122 . . . . . . . . . . . 12 (𝑡 = 𝑧 → (𝑥 ∖ {𝑡}) = (𝑥 ∖ {𝑧}))
2120unieqd 4922 . . . . . . . . . . 11 (𝑡 = 𝑧 (𝑥 ∖ {𝑡}) = (𝑥 ∖ {𝑧}))
2221difeq2d 4122 . . . . . . . . . 10 (𝑡 = 𝑧 → (𝑧 (𝑥 ∖ {𝑡})) = (𝑧 (𝑥 ∖ {𝑧})))
2318, 22eqtrd 2772 . . . . . . . . 9 (𝑡 = 𝑧 → (𝑡 (𝑥 ∖ {𝑡})) = (𝑧 (𝑥 ∖ {𝑧})))
2423ineq2d 4212 . . . . . . . 8 (𝑡 = 𝑧 → (𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡}))) = (𝑧 ∩ (𝑧 (𝑥 ∖ {𝑧}))))
2517, 24iunxsn 5094 . . . . . . 7 𝑡 ∈ {𝑧} (𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡}))) = (𝑧 ∩ (𝑧 (𝑥 ∖ {𝑧})))
2625uneq1i 4159 . . . . . 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 2760 . . . . 5 𝑡 ∈ ({𝑧} ∪ (𝑥 ∖ {𝑧}))(𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡}))) = ((𝑧 ∩ (𝑧 (𝑥 ∖ {𝑧}))) ∪ 𝑡 ∈ (𝑥 ∖ {𝑧})(𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡}))))
28 eldifsni 4793 . . . . . . . . . 10 (𝑡 ∈ (𝑥 ∖ {𝑧}) → 𝑡𝑧)
29 incom 4201 . . . . . . . . . . . 12 (𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡}))) = ((𝑡 (𝑥 ∖ {𝑡})) ∩ 𝑧)
30 kmlem4 10150 . . . . . . . . . . . 12 ((𝑧𝑥𝑡𝑧) → ((𝑡 (𝑥 ∖ {𝑡})) ∩ 𝑧) = ∅)
3129, 30eqtrid 2784 . . . . . . . . . . 11 ((𝑧𝑥𝑡𝑧) → (𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡}))) = ∅)
3231ex 413 . . . . . . . . . 10 (𝑧𝑥 → (𝑡𝑧 → (𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡}))) = ∅))
3328, 32syl5 34 . . . . . . . . 9 (𝑧𝑥 → (𝑡 ∈ (𝑥 ∖ {𝑧}) → (𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡}))) = ∅))
3433ralrimiv 3145 . . . . . . . 8 (𝑧𝑥 → ∀𝑡 ∈ (𝑥 ∖ {𝑧})(𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡}))) = ∅)
35 iuneq2 5016 . . . . . . . 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 5065 . . . . . . 7 𝑡 ∈ (𝑥 ∖ {𝑧})∅ = ∅
3836, 37eqtrdi 2788 . . . . . 6 (𝑧𝑥 𝑡 ∈ (𝑥 ∖ {𝑧})(𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡}))) = ∅)
3938uneq2d 4163 . . . . 5 (𝑧𝑥 → ((𝑧 ∩ (𝑧 (𝑥 ∖ {𝑧}))) ∪ 𝑡 ∈ (𝑥 ∖ {𝑧})(𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡})))) = ((𝑧 ∩ (𝑧 (𝑥 ∖ {𝑧}))) ∪ ∅))
4027, 39eqtrid 2784 . . . 4 (𝑧𝑥 𝑡 ∈ ({𝑧} ∪ (𝑥 ∖ {𝑧}))(𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡}))) = ((𝑧 ∩ (𝑧 (𝑥 ∖ {𝑧}))) ∪ ∅))
4115, 40eqtrd 2772 . . 3 (𝑧𝑥 𝑡𝑥 (𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡}))) = ((𝑧 ∩ (𝑧 (𝑥 ∖ {𝑧}))) ∪ ∅))
42 un0 4390 . . . 4 ((𝑧 ∩ (𝑧 (𝑥 ∖ {𝑧}))) ∪ ∅) = (𝑧 ∩ (𝑧 (𝑥 ∖ {𝑧})))
43 indif 4269 . . . 4 (𝑧 ∩ (𝑧 (𝑥 ∖ {𝑧}))) = (𝑧 (𝑥 ∖ {𝑧}))
4442, 43eqtri 2760 . . 3 ((𝑧 ∩ (𝑧 (𝑥 ∖ {𝑧}))) ∪ ∅) = (𝑧 (𝑥 ∖ {𝑧}))
4541, 44eqtrdi 2788 . 2 (𝑧𝑥 𝑡𝑥 (𝑧 ∩ (𝑡 (𝑥 ∖ {𝑡}))) = (𝑧 (𝑥 ∖ {𝑧})))
469, 45eqtrid 2784 1 (𝑧𝑥 → (𝑧 𝐴) = (𝑧 (𝑥 ∖ {𝑧})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  {cab 2709  wne 2940  wral 3061  wrex 3070  cdif 3945  cun 3946  cin 3947  wss 3948  c0 4322  {csn 4628   cuni 4908   ciun 4997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2703  ax-sep 5299
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-sn 4629  df-uni 4909  df-iun 4999
This theorem is referenced by:  kmlem12  10158
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