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Theorem dissneqlem 37341
Description: This is the core of the proof of dissneq 37342, 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 22936 . . . 4 (𝐵 ∈ (TopOn‘𝐴) → ({∅, 𝐴} ⊆ 𝐵𝐵 ⊆ 𝒫 𝐴))
21adantl 481 . . 3 ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴)) → ({∅, 𝐴} ⊆ 𝐵𝐵 ⊆ 𝒫 𝐴))
32simprd 495 . 2 ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴)) → 𝐵 ⊆ 𝒫 𝐴)
4 velpw 4605 . . . . . . 7 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
5 simp3 1139 . . . . . . . . . 10 ((𝐶𝐵𝑥𝐴𝐵 ∈ (TopOn‘𝐴)) → 𝐵 ∈ (TopOn‘𝐴))
6 df-ima 5698 . . . . . . . . . . . . . . . . . 18 ((𝑧𝐴 ↦ {𝑧}) “ 𝑥) = ran ((𝑧𝐴 ↦ {𝑧}) ↾ 𝑥)
7 resmpt 6055 . . . . . . . . . . . . . . . . . . 19 (𝑥𝐴 → ((𝑧𝐴 ↦ {𝑧}) ↾ 𝑥) = (𝑧𝑥 ↦ {𝑧}))
87rneqd 5949 . . . . . . . . . . . . . . . . . 18 (𝑥𝐴 → ran ((𝑧𝐴 ↦ {𝑧}) ↾ 𝑥) = ran (𝑧𝑥 ↦ {𝑧}))
96, 8eqtrid 2789 . . . . . . . . . . . . . . . . 17 (𝑥𝐴 → ((𝑧𝐴 ↦ {𝑧}) “ 𝑥) = ran (𝑧𝑥 ↦ {𝑧}))
10 rnmptsn 37336 . . . . . . . . . . . . . . . . 17 ran (𝑧𝑥 ↦ {𝑧}) = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}}
119, 10eqtrdi 2793 . . . . . . . . . . . . . . . 16 (𝑥𝐴 → ((𝑧𝐴 ↦ {𝑧}) “ 𝑥) = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}})
12 imassrn 6089 . . . . . . . . . . . . . . . 16 ((𝑧𝐴 ↦ {𝑧}) “ 𝑥) ⊆ ran (𝑧𝐴 ↦ {𝑧})
1311, 12eqsstrrdi 4029 . . . . . . . . . . . . . . 15 (𝑥𝐴 → {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ ran (𝑧𝐴 ↦ {𝑧}))
14 rnmptsn 37336 . . . . . . . . . . . . . . 15 ran (𝑧𝐴 ↦ {𝑧}) = {𝑢 ∣ ∃𝑧𝐴 𝑢 = {𝑧}}
1513, 14sseqtrdi 4024 . . . . . . . . . . . . . 14 (𝑥𝐴 → {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ {𝑢 ∣ ∃𝑧𝐴 𝑢 = {𝑧}})
16 dissneq.c . . . . . . . . . . . . . . 15 𝐶 = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}
17 sneq 4636 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑧 → {𝑥} = {𝑧})
1817eqeq2d 2748 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → (𝑢 = {𝑥} ↔ 𝑢 = {𝑧}))
1918cbvrexvw 3238 . . . . . . . . . . . . . . . 16 (∃𝑥𝐴 𝑢 = {𝑥} ↔ ∃𝑧𝐴 𝑢 = {𝑧})
2019abbii 2809 . . . . . . . . . . . . . . 15 {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}} = {𝑢 ∣ ∃𝑧𝐴 𝑢 = {𝑧}}
2116, 20eqtri 2765 . . . . . . . . . . . . . 14 𝐶 = {𝑢 ∣ ∃𝑧𝐴 𝑢 = {𝑧}}
2215, 21sseqtrrdi 4025 . . . . . . . . . . . . 13 (𝑥𝐴 → {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐶)
2322adantl 481 . . . . . . . . . . . 12 ((𝐶𝐵𝑥𝐴) → {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐶)
24 sstr 3992 . . . . . . . . . . . . . 14 (({𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐶𝐶𝐵) → {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐵)
2524expcom 413 . . . . . . . . . . . . 13 (𝐶𝐵 → ({𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐶 → {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐵))
2625adantr 480 . . . . . . . . . . . 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 5324 . . . . . . . . 9 ((𝐶𝐵𝑥𝐴𝐵 ∈ (TopOn‘𝐴)) → {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ∈ V)
30 isset 3494 . . . . . . . . 9 ({𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ∈ V ↔ ∃𝑦 𝑦 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}})
3129, 30sylib 218 . . . . . . . 8 ((𝐶𝐵𝑥𝐴𝐵 ∈ (TopOn‘𝐴)) → ∃𝑦 𝑦 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}})
32 eqid 2737 . . . . . . . . . . . . . . 15 (𝑧𝐴 ↦ {𝑧}) = (𝑧𝐴 ↦ {𝑧})
33 eqid 2737 . . . . . . . . . . . . . . 15 {𝑢 ∣ ∃𝑧𝐴 𝑢 = {𝑧}} = {𝑢 ∣ ∃𝑧𝐴 𝑢 = {𝑧}}
3432, 33mptsnun 37340 . . . . . . . . . . . . . 14 (𝑥𝐴𝑥 = ((𝑧𝐴 ↦ {𝑧}) “ 𝑥))
3511unieqd 4920 . . . . . . . . . . . . . 14 (𝑥𝐴 ((𝑧𝐴 ↦ {𝑧}) “ 𝑥) = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}})
3634, 35eqtrd 2777 . . . . . . . . . . . . 13 (𝑥𝐴𝑥 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}})
3736adantl 481 . . . . . . . . . . . 12 ((𝐶𝐵𝑥𝐴) → 𝑥 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}})
3827, 37jca 511 . . . . . . . . . . 11 ((𝐶𝐵𝑥𝐴) → ({𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐵𝑥 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}}))
39 sseq1 4009 . . . . . . . . . . . 12 (𝑦 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} → (𝑦𝐵 ↔ {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐵))
40 unieq 4918 . . . . . . . . . . . . 13 (𝑦 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} → 𝑦 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}})
4140eqeq2d 2748 . . . . . . . . . . . 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 247 . . . . . . . . . 10 ((𝐶𝐵𝑥𝐴) → (𝑦 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} → (𝑦𝐵𝑥 = 𝑦)))
4443eximdv 1917 . . . . . . . . 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 1406 . . . . . 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 22919 . . . . . . . 8 (𝐵 ∈ (TopOn‘𝐴) → 𝐵 ∈ Top)
51 tgtop 22980 . . . . . . . 8 (𝐵 ∈ Top → (topGen‘𝐵) = 𝐵)
5250, 51syl 17 . . . . . . 7 (𝐵 ∈ (TopOn‘𝐴) → (topGen‘𝐵) = 𝐵)
5352eleq2d 2827 . . . . . 6 (𝐵 ∈ (TopOn‘𝐴) → (𝑥 ∈ (topGen‘𝐵) ↔ 𝑥𝐵))
54 eltg3 22969 . . . . . 6 (𝐵 ∈ (TopOn‘𝐴) → (𝑥 ∈ (topGen‘𝐵) ↔ ∃𝑦(𝑦𝐵𝑥 = 𝑦)))
5553, 54bitr3d 281 . . . . 5 (𝐵 ∈ (TopOn‘𝐴) → (𝑥𝐵 ↔ ∃𝑦(𝑦𝐵𝑥 = 𝑦)))
5655adantl 481 . . . 4 ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴)) → (𝑥𝐵 ↔ ∃𝑦(𝑦𝐵𝑥 = 𝑦)))
5749, 56sylibrd 259 . . 3 ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴)) → (𝑥 ∈ 𝒫 𝐴𝑥𝐵))
5857ssrdv 3989 . 2 ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴)) → 𝒫 𝐴𝐵)
593, 58eqssd 4001 1 ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴)) → 𝐵 = 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wex 1779  wcel 2108  {cab 2714  wrex 3070  Vcvv 3480  wss 3951  c0 4333  𝒫 cpw 4600  {csn 4626  {cpr 4628   cuni 4907  cmpt 5225  ran crn 5686  cres 5687  cima 5688  cfv 6561  topGenctg 17482  Topctop 22899  TopOnctopon 22916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569  df-topgen 17488  df-top 22900  df-topon 22917
This theorem is referenced by:  dissneq  37342
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