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Theorem dissneqlem 36126
Description: This is the core of the proof of dissneq 36127, but to avoid the distinct variables on the definitions, we split this proof into two. (Contributed by ML, 16-Jul-2020.)
Hypothesis
Ref Expression
dissneq.c 𝐶 = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}
Assertion
Ref Expression
dissneqlem ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴)) → 𝐵 = 𝒫 𝐴)
Distinct variable groups:   𝑢,𝐴,𝑥   𝑥,𝐵   𝑥,𝐶
Allowed substitution hints:   𝐵(𝑢)   𝐶(𝑢)

Proof of Theorem dissneqlem
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 topgele 22401 . . . 4 (𝐵 ∈ (TopOn‘𝐴) → ({∅, 𝐴} ⊆ 𝐵𝐵 ⊆ 𝒫 𝐴))
21adantl 483 . . 3 ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴)) → ({∅, 𝐴} ⊆ 𝐵𝐵 ⊆ 𝒫 𝐴))
32simprd 497 . 2 ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴)) → 𝐵 ⊆ 𝒫 𝐴)
4 velpw 4603 . . . . . . 7 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
5 simp3 1139 . . . . . . . . . 10 ((𝐶𝐵𝑥𝐴𝐵 ∈ (TopOn‘𝐴)) → 𝐵 ∈ (TopOn‘𝐴))
6 df-ima 5685 . . . . . . . . . . . . . . . . . 18 ((𝑧𝐴 ↦ {𝑧}) “ 𝑥) = ran ((𝑧𝐴 ↦ {𝑧}) ↾ 𝑥)
7 resmpt 6030 . . . . . . . . . . . . . . . . . . 19 (𝑥𝐴 → ((𝑧𝐴 ↦ {𝑧}) ↾ 𝑥) = (𝑧𝑥 ↦ {𝑧}))
87rneqd 5932 . . . . . . . . . . . . . . . . . 18 (𝑥𝐴 → ran ((𝑧𝐴 ↦ {𝑧}) ↾ 𝑥) = ran (𝑧𝑥 ↦ {𝑧}))
96, 8eqtrid 2785 . . . . . . . . . . . . . . . . 17 (𝑥𝐴 → ((𝑧𝐴 ↦ {𝑧}) “ 𝑥) = ran (𝑧𝑥 ↦ {𝑧}))
10 rnmptsn 36121 . . . . . . . . . . . . . . . . 17 ran (𝑧𝑥 ↦ {𝑧}) = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}}
119, 10eqtrdi 2789 . . . . . . . . . . . . . . . 16 (𝑥𝐴 → ((𝑧𝐴 ↦ {𝑧}) “ 𝑥) = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}})
12 imassrn 6063 . . . . . . . . . . . . . . . 16 ((𝑧𝐴 ↦ {𝑧}) “ 𝑥) ⊆ ran (𝑧𝐴 ↦ {𝑧})
1311, 12eqsstrrdi 4035 . . . . . . . . . . . . . . 15 (𝑥𝐴 → {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ ran (𝑧𝐴 ↦ {𝑧}))
14 rnmptsn 36121 . . . . . . . . . . . . . . 15 ran (𝑧𝐴 ↦ {𝑧}) = {𝑢 ∣ ∃𝑧𝐴 𝑢 = {𝑧}}
1513, 14sseqtrdi 4030 . . . . . . . . . . . . . 14 (𝑥𝐴 → {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ {𝑢 ∣ ∃𝑧𝐴 𝑢 = {𝑧}})
16 dissneq.c . . . . . . . . . . . . . . 15 𝐶 = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}
17 sneq 4634 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑧 → {𝑥} = {𝑧})
1817eqeq2d 2744 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → (𝑢 = {𝑥} ↔ 𝑢 = {𝑧}))
1918cbvrexvw 3236 . . . . . . . . . . . . . . . 16 (∃𝑥𝐴 𝑢 = {𝑥} ↔ ∃𝑧𝐴 𝑢 = {𝑧})
2019abbii 2803 . . . . . . . . . . . . . . 15 {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}} = {𝑢 ∣ ∃𝑧𝐴 𝑢 = {𝑧}}
2116, 20eqtri 2761 . . . . . . . . . . . . . 14 𝐶 = {𝑢 ∣ ∃𝑧𝐴 𝑢 = {𝑧}}
2215, 21sseqtrrdi 4031 . . . . . . . . . . . . 13 (𝑥𝐴 → {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐶)
2322adantl 483 . . . . . . . . . . . 12 ((𝐶𝐵𝑥𝐴) → {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐶)
24 sstr 3988 . . . . . . . . . . . . . 14 (({𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐶𝐶𝐵) → {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐵)
2524expcom 415 . . . . . . . . . . . . 13 (𝐶𝐵 → ({𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐶 → {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐵))
2625adantr 482 . . . . . . . . . . . 12 ((𝐶𝐵𝑥𝐴) → ({𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐶 → {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐵))
2723, 26mpd 15 . . . . . . . . . . 11 ((𝐶𝐵𝑥𝐴) → {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐵)
28273adant3 1133 . . . . . . . . . 10 ((𝐶𝐵𝑥𝐴𝐵 ∈ (TopOn‘𝐴)) → {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐵)
295, 28ssexd 5320 . . . . . . . . 9 ((𝐶𝐵𝑥𝐴𝐵 ∈ (TopOn‘𝐴)) → {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ∈ V)
30 isset 3488 . . . . . . . . 9 ({𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ∈ V ↔ ∃𝑦 𝑦 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}})
3129, 30sylib 217 . . . . . . . 8 ((𝐶𝐵𝑥𝐴𝐵 ∈ (TopOn‘𝐴)) → ∃𝑦 𝑦 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}})
32 eqid 2733 . . . . . . . . . . . . . . 15 (𝑧𝐴 ↦ {𝑧}) = (𝑧𝐴 ↦ {𝑧})
33 eqid 2733 . . . . . . . . . . . . . . 15 {𝑢 ∣ ∃𝑧𝐴 𝑢 = {𝑧}} = {𝑢 ∣ ∃𝑧𝐴 𝑢 = {𝑧}}
3432, 33mptsnun 36125 . . . . . . . . . . . . . 14 (𝑥𝐴𝑥 = ((𝑧𝐴 ↦ {𝑧}) “ 𝑥))
3511unieqd 4918 . . . . . . . . . . . . . 14 (𝑥𝐴 ((𝑧𝐴 ↦ {𝑧}) “ 𝑥) = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}})
3634, 35eqtrd 2773 . . . . . . . . . . . . 13 (𝑥𝐴𝑥 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}})
3736adantl 483 . . . . . . . . . . . 12 ((𝐶𝐵𝑥𝐴) → 𝑥 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}})
3827, 37jca 513 . . . . . . . . . . 11 ((𝐶𝐵𝑥𝐴) → ({𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐵𝑥 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}}))
39 sseq1 4005 . . . . . . . . . . . 12 (𝑦 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} → (𝑦𝐵 ↔ {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐵))
40 unieq 4915 . . . . . . . . . . . . 13 (𝑦 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} → 𝑦 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}})
4140eqeq2d 2744 . . . . . . . . . . . 12 (𝑦 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} → (𝑥 = 𝑦𝑥 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}}))
4239, 41anbi12d 632 . . . . . . . . . . 11 (𝑦 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} → ((𝑦𝐵𝑥 = 𝑦) ↔ ({𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐵𝑥 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}})))
4338, 42syl5ibrcom 246 . . . . . . . . . 10 ((𝐶𝐵𝑥𝐴) → (𝑦 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} → (𝑦𝐵𝑥 = 𝑦)))
4443eximdv 1921 . . . . . . . . 9 ((𝐶𝐵𝑥𝐴) → (∃𝑦 𝑦 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} → ∃𝑦(𝑦𝐵𝑥 = 𝑦)))
45443adant3 1133 . . . . . . . 8 ((𝐶𝐵𝑥𝐴𝐵 ∈ (TopOn‘𝐴)) → (∃𝑦 𝑦 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} → ∃𝑦(𝑦𝐵𝑥 = 𝑦)))
4631, 45mpd 15 . . . . . . 7 ((𝐶𝐵𝑥𝐴𝐵 ∈ (TopOn‘𝐴)) → ∃𝑦(𝑦𝐵𝑥 = 𝑦))
474, 46syl3an2b 1405 . . . . . 6 ((𝐶𝐵𝑥 ∈ 𝒫 𝐴𝐵 ∈ (TopOn‘𝐴)) → ∃𝑦(𝑦𝐵𝑥 = 𝑦))
48473com23 1127 . . . . 5 ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴) ∧ 𝑥 ∈ 𝒫 𝐴) → ∃𝑦(𝑦𝐵𝑥 = 𝑦))
49483expia 1122 . . . 4 ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴)) → (𝑥 ∈ 𝒫 𝐴 → ∃𝑦(𝑦𝐵𝑥 = 𝑦)))
50 topontop 22384 . . . . . . . 8 (𝐵 ∈ (TopOn‘𝐴) → 𝐵 ∈ Top)
51 tgtop 22445 . . . . . . . 8 (𝐵 ∈ Top → (topGen‘𝐵) = 𝐵)
5250, 51syl 17 . . . . . . 7 (𝐵 ∈ (TopOn‘𝐴) → (topGen‘𝐵) = 𝐵)
5352eleq2d 2820 . . . . . 6 (𝐵 ∈ (TopOn‘𝐴) → (𝑥 ∈ (topGen‘𝐵) ↔ 𝑥𝐵))
54 eltg3 22434 . . . . . 6 (𝐵 ∈ (TopOn‘𝐴) → (𝑥 ∈ (topGen‘𝐵) ↔ ∃𝑦(𝑦𝐵𝑥 = 𝑦)))
5553, 54bitr3d 281 . . . . 5 (𝐵 ∈ (TopOn‘𝐴) → (𝑥𝐵 ↔ ∃𝑦(𝑦𝐵𝑥 = 𝑦)))
5655adantl 483 . . . 4 ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴)) → (𝑥𝐵 ↔ ∃𝑦(𝑦𝐵𝑥 = 𝑦)))
5749, 56sylibrd 259 . . 3 ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴)) → (𝑥 ∈ 𝒫 𝐴𝑥𝐵))
5857ssrdv 3986 . 2 ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴)) → 𝒫 𝐴𝐵)
593, 58eqssd 3997 1 ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴)) → 𝐵 = 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wex 1782  wcel 2107  {cab 2710  wrex 3071  Vcvv 3475  wss 3946  c0 4320  𝒫 cpw 4598  {csn 4624  {cpr 4626   cuni 4904  cmpt 5227  ran crn 5673  cres 5674  cima 5675  cfv 6535  topGenctg 17370  Topctop 22364  TopOnctopon 22381
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-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423  ax-un 7712
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3776  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6487  df-fun 6537  df-fv 6543  df-topgen 17376  df-top 22365  df-topon 22382
This theorem is referenced by:  dissneq  36127
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