Users' Mathboxes Mathbox for ML < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dissneqlem Structured version   Visualization version   GIF version

Theorem dissneqlem 33490
Description: This is the core of the proof of dissneq 33491, 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 20928 . . . 4 (𝐵 ∈ (TopOn‘𝐴) → ({∅, 𝐴} ⊆ 𝐵𝐵 ⊆ 𝒫 𝐴))
21adantl 473 . . 3 ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴)) → ({∅, 𝐴} ⊆ 𝐵𝐵 ⊆ 𝒫 𝐴))
32simprd 482 . 2 ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴)) → 𝐵 ⊆ 𝒫 𝐴)
4 selpw 4301 . . . . . . 7 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
5 simp3 1132 . . . . . . . . . 10 ((𝐶𝐵𝑥𝐴𝐵 ∈ (TopOn‘𝐴)) → 𝐵 ∈ (TopOn‘𝐴))
6 df-ima 5271 . . . . . . . . . . . . . . . . . 18 ((𝑧𝐴 ↦ {𝑧}) “ 𝑥) = ran ((𝑧𝐴 ↦ {𝑧}) ↾ 𝑥)
7 resmpt 5599 . . . . . . . . . . . . . . . . . . 19 (𝑥𝐴 → ((𝑧𝐴 ↦ {𝑧}) ↾ 𝑥) = (𝑧𝑥 ↦ {𝑧}))
87rneqd 5500 . . . . . . . . . . . . . . . . . 18 (𝑥𝐴 → ran ((𝑧𝐴 ↦ {𝑧}) ↾ 𝑥) = ran (𝑧𝑥 ↦ {𝑧}))
96, 8syl5eq 2798 . . . . . . . . . . . . . . . . 17 (𝑥𝐴 → ((𝑧𝐴 ↦ {𝑧}) “ 𝑥) = ran (𝑧𝑥 ↦ {𝑧}))
10 rnmptsn 33485 . . . . . . . . . . . . . . . . 17 ran (𝑧𝑥 ↦ {𝑧}) = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}}
119, 10syl6eq 2802 . . . . . . . . . . . . . . . 16 (𝑥𝐴 → ((𝑧𝐴 ↦ {𝑧}) “ 𝑥) = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}})
12 imassrn 5627 . . . . . . . . . . . . . . . 16 ((𝑧𝐴 ↦ {𝑧}) “ 𝑥) ⊆ ran (𝑧𝐴 ↦ {𝑧})
1311, 12syl6eqssr 3789 . . . . . . . . . . . . . . 15 (𝑥𝐴 → {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ ran (𝑧𝐴 ↦ {𝑧}))
14 rnmptsn 33485 . . . . . . . . . . . . . . 15 ran (𝑧𝐴 ↦ {𝑧}) = {𝑢 ∣ ∃𝑧𝐴 𝑢 = {𝑧}}
1513, 14syl6sseq 3784 . . . . . . . . . . . . . 14 (𝑥𝐴 → {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ {𝑢 ∣ ∃𝑧𝐴 𝑢 = {𝑧}})
16 dissneq.c . . . . . . . . . . . . . . 15 𝐶 = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}
17 sneq 4323 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑧 → {𝑥} = {𝑧})
1817eqeq2d 2762 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → (𝑢 = {𝑥} ↔ 𝑢 = {𝑧}))
1918cbvrexv 3303 . . . . . . . . . . . . . . . 16 (∃𝑥𝐴 𝑢 = {𝑥} ↔ ∃𝑧𝐴 𝑢 = {𝑧})
2019abbii 2869 . . . . . . . . . . . . . . 15 {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}} = {𝑢 ∣ ∃𝑧𝐴 𝑢 = {𝑧}}
2116, 20eqtri 2774 . . . . . . . . . . . . . 14 𝐶 = {𝑢 ∣ ∃𝑧𝐴 𝑢 = {𝑧}}
2215, 21syl6sseqr 3785 . . . . . . . . . . . . 13 (𝑥𝐴 → {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐶)
2322adantl 473 . . . . . . . . . . . 12 ((𝐶𝐵𝑥𝐴) → {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐶)
24 sstr 3744 . . . . . . . . . . . . . 14 (({𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐶𝐶𝐵) → {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐵)
2524expcom 450 . . . . . . . . . . . . 13 (𝐶𝐵 → ({𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐶 → {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐵))
2625adantr 472 . . . . . . . . . . . 12 ((𝐶𝐵𝑥𝐴) → ({𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐶 → {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐵))
2723, 26mpd 15 . . . . . . . . . . 11 ((𝐶𝐵𝑥𝐴) → {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐵)
28273adant3 1126 . . . . . . . . . 10 ((𝐶𝐵𝑥𝐴𝐵 ∈ (TopOn‘𝐴)) → {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐵)
295, 28ssexd 4949 . . . . . . . . 9 ((𝐶𝐵𝑥𝐴𝐵 ∈ (TopOn‘𝐴)) → {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ∈ V)
30 isset 3339 . . . . . . . . 9 ({𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ∈ V ↔ ∃𝑦 𝑦 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}})
3129, 30sylib 208 . . . . . . . 8 ((𝐶𝐵𝑥𝐴𝐵 ∈ (TopOn‘𝐴)) → ∃𝑦 𝑦 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}})
32 eqid 2752 . . . . . . . . . . . . . . 15 (𝑧𝐴 ↦ {𝑧}) = (𝑧𝐴 ↦ {𝑧})
33 eqid 2752 . . . . . . . . . . . . . . 15 {𝑢 ∣ ∃𝑧𝐴 𝑢 = {𝑧}} = {𝑢 ∣ ∃𝑧𝐴 𝑢 = {𝑧}}
3432, 33mptsnun 33489 . . . . . . . . . . . . . 14 (𝑥𝐴𝑥 = ((𝑧𝐴 ↦ {𝑧}) “ 𝑥))
3511unieqd 4590 . . . . . . . . . . . . . 14 (𝑥𝐴 ((𝑧𝐴 ↦ {𝑧}) “ 𝑥) = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}})
3634, 35eqtrd 2786 . . . . . . . . . . . . 13 (𝑥𝐴𝑥 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}})
3736adantl 473 . . . . . . . . . . . 12 ((𝐶𝐵𝑥𝐴) → 𝑥 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}})
3827, 37jca 555 . . . . . . . . . . 11 ((𝐶𝐵𝑥𝐴) → ({𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐵𝑥 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}}))
39 sseq1 3759 . . . . . . . . . . . 12 (𝑦 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} → (𝑦𝐵 ↔ {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐵))
40 unieq 4588 . . . . . . . . . . . . 13 (𝑦 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} → 𝑦 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}})
4140eqeq2d 2762 . . . . . . . . . . . 12 (𝑦 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} → (𝑥 = 𝑦𝑥 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}}))
4239, 41anbi12d 749 . . . . . . . . . . 11 (𝑦 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} → ((𝑦𝐵𝑥 = 𝑦) ↔ ({𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐵𝑥 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}})))
4338, 42syl5ibrcom 237 . . . . . . . . . 10 ((𝐶𝐵𝑥𝐴) → (𝑦 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} → (𝑦𝐵𝑥 = 𝑦)))
4443eximdv 1987 . . . . . . . . 9 ((𝐶𝐵𝑥𝐴) → (∃𝑦 𝑦 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} → ∃𝑦(𝑦𝐵𝑥 = 𝑦)))
45443adant3 1126 . . . . . . . 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 1507 . . . . . 6 ((𝐶𝐵𝑥 ∈ 𝒫 𝐴𝐵 ∈ (TopOn‘𝐴)) → ∃𝑦(𝑦𝐵𝑥 = 𝑦))
48473com23 1120 . . . . 5 ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴) ∧ 𝑥 ∈ 𝒫 𝐴) → ∃𝑦(𝑦𝐵𝑥 = 𝑦))
49483expia 1114 . . . 4 ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴)) → (𝑥 ∈ 𝒫 𝐴 → ∃𝑦(𝑦𝐵𝑥 = 𝑦)))
50 topontop 20912 . . . . . . . 8 (𝐵 ∈ (TopOn‘𝐴) → 𝐵 ∈ Top)
51 tgtop 20971 . . . . . . . 8 (𝐵 ∈ Top → (topGen‘𝐵) = 𝐵)
5250, 51syl 17 . . . . . . 7 (𝐵 ∈ (TopOn‘𝐴) → (topGen‘𝐵) = 𝐵)
5352eleq2d 2817 . . . . . 6 (𝐵 ∈ (TopOn‘𝐴) → (𝑥 ∈ (topGen‘𝐵) ↔ 𝑥𝐵))
54 eltg3 20960 . . . . . 6 (𝐵 ∈ (TopOn‘𝐴) → (𝑥 ∈ (topGen‘𝐵) ↔ ∃𝑦(𝑦𝐵𝑥 = 𝑦)))
5553, 54bitr3d 270 . . . . 5 (𝐵 ∈ (TopOn‘𝐴) → (𝑥𝐵 ↔ ∃𝑦(𝑦𝐵𝑥 = 𝑦)))
5655adantl 473 . . . 4 ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴)) → (𝑥𝐵 ↔ ∃𝑦(𝑦𝐵𝑥 = 𝑦)))
5749, 56sylibrd 249 . . 3 ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴)) → (𝑥 ∈ 𝒫 𝐴𝑥𝐵))
5857ssrdv 3742 . 2 ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴)) → 𝒫 𝐴𝐵)
593, 58eqssd 3753 1 ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴)) → 𝐵 = 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1624  wex 1845  wcel 2131  {cab 2738  wrex 3043  Vcvv 3332  wss 3707  c0 4050  𝒫 cpw 4294  {csn 4313  {cpr 4315   cuni 4580  cmpt 4873  ran crn 5259  cres 5260  cima 5261  cfv 6041  topGenctg 16292  Topctop 20892  TopOnctopon 20909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fv 6049  df-topgen 16298  df-top 20893  df-topon 20910
This theorem is referenced by:  dissneq  33491
  Copyright terms: Public domain W3C validator