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Theorem dissneqlem 33642
Description: This is the core of the proof of dissneq 33643, 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 21028 . . . 4 (𝐵 ∈ (TopOn‘𝐴) → ({∅, 𝐴} ⊆ 𝐵𝐵 ⊆ 𝒫 𝐴))
21adantl 473 . . 3 ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴)) → ({∅, 𝐴} ⊆ 𝐵𝐵 ⊆ 𝒫 𝐴))
32simprd 489 . 2 ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴)) → 𝐵 ⊆ 𝒫 𝐴)
4 selpw 4324 . . . . . . 7 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
5 simp3 1168 . . . . . . . . . 10 ((𝐶𝐵𝑥𝐴𝐵 ∈ (TopOn‘𝐴)) → 𝐵 ∈ (TopOn‘𝐴))
6 df-ima 5292 . . . . . . . . . . . . . . . . . 18 ((𝑧𝐴 ↦ {𝑧}) “ 𝑥) = ran ((𝑧𝐴 ↦ {𝑧}) ↾ 𝑥)
7 resmpt 5628 . . . . . . . . . . . . . . . . . . 19 (𝑥𝐴 → ((𝑧𝐴 ↦ {𝑧}) ↾ 𝑥) = (𝑧𝑥 ↦ {𝑧}))
87rneqd 5523 . . . . . . . . . . . . . . . . . 18 (𝑥𝐴 → ran ((𝑧𝐴 ↦ {𝑧}) ↾ 𝑥) = ran (𝑧𝑥 ↦ {𝑧}))
96, 8syl5eq 2811 . . . . . . . . . . . . . . . . 17 (𝑥𝐴 → ((𝑧𝐴 ↦ {𝑧}) “ 𝑥) = ran (𝑧𝑥 ↦ {𝑧}))
10 rnmptsn 33637 . . . . . . . . . . . . . . . . 17 ran (𝑧𝑥 ↦ {𝑧}) = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}}
119, 10syl6eq 2815 . . . . . . . . . . . . . . . 16 (𝑥𝐴 → ((𝑧𝐴 ↦ {𝑧}) “ 𝑥) = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}})
12 imassrn 5661 . . . . . . . . . . . . . . . 16 ((𝑧𝐴 ↦ {𝑧}) “ 𝑥) ⊆ ran (𝑧𝐴 ↦ {𝑧})
1311, 12syl6eqssr 3818 . . . . . . . . . . . . . . 15 (𝑥𝐴 → {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ ran (𝑧𝐴 ↦ {𝑧}))
14 rnmptsn 33637 . . . . . . . . . . . . . . 15 ran (𝑧𝐴 ↦ {𝑧}) = {𝑢 ∣ ∃𝑧𝐴 𝑢 = {𝑧}}
1513, 14syl6sseq 3813 . . . . . . . . . . . . . 14 (𝑥𝐴 → {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ {𝑢 ∣ ∃𝑧𝐴 𝑢 = {𝑧}})
16 dissneq.c . . . . . . . . . . . . . . 15 𝐶 = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}
17 sneq 4346 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑧 → {𝑥} = {𝑧})
1817eqeq2d 2775 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → (𝑢 = {𝑥} ↔ 𝑢 = {𝑧}))
1918cbvrexv 3320 . . . . . . . . . . . . . . . 16 (∃𝑥𝐴 𝑢 = {𝑥} ↔ ∃𝑧𝐴 𝑢 = {𝑧})
2019abbii 2882 . . . . . . . . . . . . . . 15 {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}} = {𝑢 ∣ ∃𝑧𝐴 𝑢 = {𝑧}}
2116, 20eqtri 2787 . . . . . . . . . . . . . 14 𝐶 = {𝑢 ∣ ∃𝑧𝐴 𝑢 = {𝑧}}
2215, 21syl6sseqr 3814 . . . . . . . . . . . . 13 (𝑥𝐴 → {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐶)
2322adantl 473 . . . . . . . . . . . 12 ((𝐶𝐵𝑥𝐴) → {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐶)
24 sstr 3771 . . . . . . . . . . . . . 14 (({𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐶𝐶𝐵) → {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐵)
2524expcom 402 . . . . . . . . . . . . 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 1162 . . . . . . . . . 10 ((𝐶𝐵𝑥𝐴𝐵 ∈ (TopOn‘𝐴)) → {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐵)
295, 28ssexd 4968 . . . . . . . . 9 ((𝐶𝐵𝑥𝐴𝐵 ∈ (TopOn‘𝐴)) → {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ∈ V)
30 isset 3360 . . . . . . . . 9 ({𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ∈ V ↔ ∃𝑦 𝑦 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}})
3129, 30sylib 209 . . . . . . . 8 ((𝐶𝐵𝑥𝐴𝐵 ∈ (TopOn‘𝐴)) → ∃𝑦 𝑦 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}})
32 eqid 2765 . . . . . . . . . . . . . . 15 (𝑧𝐴 ↦ {𝑧}) = (𝑧𝐴 ↦ {𝑧})
33 eqid 2765 . . . . . . . . . . . . . . 15 {𝑢 ∣ ∃𝑧𝐴 𝑢 = {𝑧}} = {𝑢 ∣ ∃𝑧𝐴 𝑢 = {𝑧}}
3432, 33mptsnun 33641 . . . . . . . . . . . . . 14 (𝑥𝐴𝑥 = ((𝑧𝐴 ↦ {𝑧}) “ 𝑥))
3511unieqd 4606 . . . . . . . . . . . . . 14 (𝑥𝐴 ((𝑧𝐴 ↦ {𝑧}) “ 𝑥) = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}})
3634, 35eqtrd 2799 . . . . . . . . . . . . 13 (𝑥𝐴𝑥 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}})
3736adantl 473 . . . . . . . . . . . 12 ((𝐶𝐵𝑥𝐴) → 𝑥 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}})
3827, 37jca 507 . . . . . . . . . . 11 ((𝐶𝐵𝑥𝐴) → ({𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐵𝑥 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}}))
39 sseq1 3788 . . . . . . . . . . . 12 (𝑦 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} → (𝑦𝐵 ↔ {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐵))
40 unieq 4604 . . . . . . . . . . . . 13 (𝑦 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} → 𝑦 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}})
4140eqeq2d 2775 . . . . . . . . . . . 12 (𝑦 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} → (𝑥 = 𝑦𝑥 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}}))
4239, 41anbi12d 624 . . . . . . . . . . 11 (𝑦 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} → ((𝑦𝐵𝑥 = 𝑦) ↔ ({𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐵𝑥 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}})))
4338, 42syl5ibrcom 238 . . . . . . . . . 10 ((𝐶𝐵𝑥𝐴) → (𝑦 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} → (𝑦𝐵𝑥 = 𝑦)))
4443eximdv 2012 . . . . . . . . 9 ((𝐶𝐵𝑥𝐴) → (∃𝑦 𝑦 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} → ∃𝑦(𝑦𝐵𝑥 = 𝑦)))
45443adant3 1162 . . . . . . . 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 1523 . . . . . 6 ((𝐶𝐵𝑥 ∈ 𝒫 𝐴𝐵 ∈ (TopOn‘𝐴)) → ∃𝑦(𝑦𝐵𝑥 = 𝑦))
48473com23 1156 . . . . 5 ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴) ∧ 𝑥 ∈ 𝒫 𝐴) → ∃𝑦(𝑦𝐵𝑥 = 𝑦))
49483expia 1150 . . . 4 ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴)) → (𝑥 ∈ 𝒫 𝐴 → ∃𝑦(𝑦𝐵𝑥 = 𝑦)))
50 topontop 21011 . . . . . . . 8 (𝐵 ∈ (TopOn‘𝐴) → 𝐵 ∈ Top)
51 tgtop 21071 . . . . . . . 8 (𝐵 ∈ Top → (topGen‘𝐵) = 𝐵)
5250, 51syl 17 . . . . . . 7 (𝐵 ∈ (TopOn‘𝐴) → (topGen‘𝐵) = 𝐵)
5352eleq2d 2830 . . . . . 6 (𝐵 ∈ (TopOn‘𝐴) → (𝑥 ∈ (topGen‘𝐵) ↔ 𝑥𝐵))
54 eltg3 21060 . . . . . 6 (𝐵 ∈ (TopOn‘𝐴) → (𝑥 ∈ (topGen‘𝐵) ↔ ∃𝑦(𝑦𝐵𝑥 = 𝑦)))
5553, 54bitr3d 272 . . . . 5 (𝐵 ∈ (TopOn‘𝐴) → (𝑥𝐵 ↔ ∃𝑦(𝑦𝐵𝑥 = 𝑦)))
5655adantl 473 . . . 4 ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴)) → (𝑥𝐵 ↔ ∃𝑦(𝑦𝐵𝑥 = 𝑦)))
5749, 56sylibrd 250 . . 3 ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴)) → (𝑥 ∈ 𝒫 𝐴𝑥𝐵))
5857ssrdv 3769 . 2 ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴)) → 𝒫 𝐴𝐵)
593, 58eqssd 3780 1 ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴)) → 𝐵 = 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wex 1874  wcel 2155  {cab 2751  wrex 3056  Vcvv 3350  wss 3734  c0 4081  𝒫 cpw 4317  {csn 4336  {cpr 4338   cuni 4596  cmpt 4890  ran crn 5280  cres 5281  cima 5282  cfv 6070  topGenctg 16378  Topctop 20991  TopOnctopon 21008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-iota 6033  df-fun 6072  df-fv 6078  df-topgen 16384  df-top 20992  df-topon 21009
This theorem is referenced by:  dissneq  33643
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