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Theorem dissneqlem 32819
 Description: This is the core of the proof of dissneq 32820, 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 20649 . . . 4 (𝐵 ∈ (TopOn‘𝐴) → ({∅, 𝐴} ⊆ 𝐵𝐵 ⊆ 𝒫 𝐴))
21adantl 482 . . 3 ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴)) → ({∅, 𝐴} ⊆ 𝐵𝐵 ⊆ 𝒫 𝐴))
32simprd 479 . 2 ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴)) → 𝐵 ⊆ 𝒫 𝐴)
4 selpw 4137 . . . . . . 7 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
5 simp3 1061 . . . . . . . . . 10 ((𝐶𝐵𝑥𝐴𝐵 ∈ (TopOn‘𝐴)) → 𝐵 ∈ (TopOn‘𝐴))
6 df-ima 5087 . . . . . . . . . . . . . . . . . 18 ((𝑧𝐴 ↦ {𝑧}) “ 𝑥) = ran ((𝑧𝐴 ↦ {𝑧}) ↾ 𝑥)
7 resmpt 5408 . . . . . . . . . . . . . . . . . . 19 (𝑥𝐴 → ((𝑧𝐴 ↦ {𝑧}) ↾ 𝑥) = (𝑧𝑥 ↦ {𝑧}))
87rneqd 5313 . . . . . . . . . . . . . . . . . 18 (𝑥𝐴 → ran ((𝑧𝐴 ↦ {𝑧}) ↾ 𝑥) = ran (𝑧𝑥 ↦ {𝑧}))
96, 8syl5eq 2667 . . . . . . . . . . . . . . . . 17 (𝑥𝐴 → ((𝑧𝐴 ↦ {𝑧}) “ 𝑥) = ran (𝑧𝑥 ↦ {𝑧}))
10 rnmptsn 32814 . . . . . . . . . . . . . . . . 17 ran (𝑧𝑥 ↦ {𝑧}) = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}}
119, 10syl6eq 2671 . . . . . . . . . . . . . . . 16 (𝑥𝐴 → ((𝑧𝐴 ↦ {𝑧}) “ 𝑥) = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}})
12 imassrn 5436 . . . . . . . . . . . . . . . 16 ((𝑧𝐴 ↦ {𝑧}) “ 𝑥) ⊆ ran (𝑧𝐴 ↦ {𝑧})
1311, 12syl6eqssr 3635 . . . . . . . . . . . . . . 15 (𝑥𝐴 → {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ ran (𝑧𝐴 ↦ {𝑧}))
14 rnmptsn 32814 . . . . . . . . . . . . . . 15 ran (𝑧𝐴 ↦ {𝑧}) = {𝑢 ∣ ∃𝑧𝐴 𝑢 = {𝑧}}
1513, 14syl6sseq 3630 . . . . . . . . . . . . . 14 (𝑥𝐴 → {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ {𝑢 ∣ ∃𝑧𝐴 𝑢 = {𝑧}})
16 dissneq.c . . . . . . . . . . . . . . 15 𝐶 = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}
17 sneq 4158 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑧 → {𝑥} = {𝑧})
1817eqeq2d 2631 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → (𝑢 = {𝑥} ↔ 𝑢 = {𝑧}))
1918cbvrexv 3160 . . . . . . . . . . . . . . . 16 (∃𝑥𝐴 𝑢 = {𝑥} ↔ ∃𝑧𝐴 𝑢 = {𝑧})
2019abbii 2736 . . . . . . . . . . . . . . 15 {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}} = {𝑢 ∣ ∃𝑧𝐴 𝑢 = {𝑧}}
2116, 20eqtri 2643 . . . . . . . . . . . . . 14 𝐶 = {𝑢 ∣ ∃𝑧𝐴 𝑢 = {𝑧}}
2215, 21syl6sseqr 3631 . . . . . . . . . . . . 13 (𝑥𝐴 → {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐶)
2322adantl 482 . . . . . . . . . . . 12 ((𝐶𝐵𝑥𝐴) → {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐶)
24 sstr 3591 . . . . . . . . . . . . . 14 (({𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐶𝐶𝐵) → {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐵)
2524expcom 451 . . . . . . . . . . . . 13 (𝐶𝐵 → ({𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐶 → {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐵))
2625adantr 481 . . . . . . . . . . . 12 ((𝐶𝐵𝑥𝐴) → ({𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐶 → {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐵))
2723, 26mpd 15 . . . . . . . . . . 11 ((𝐶𝐵𝑥𝐴) → {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐵)
28273adant3 1079 . . . . . . . . . 10 ((𝐶𝐵𝑥𝐴𝐵 ∈ (TopOn‘𝐴)) → {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐵)
295, 28ssexd 4765 . . . . . . . . 9 ((𝐶𝐵𝑥𝐴𝐵 ∈ (TopOn‘𝐴)) → {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ∈ V)
30 isset 3193 . . . . . . . . 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 2621 . . . . . . . . . . . . . . 15 (𝑧𝐴 ↦ {𝑧}) = (𝑧𝐴 ↦ {𝑧})
33 eqid 2621 . . . . . . . . . . . . . . 15 {𝑢 ∣ ∃𝑧𝐴 𝑢 = {𝑧}} = {𝑢 ∣ ∃𝑧𝐴 𝑢 = {𝑧}}
3432, 33mptsnun 32818 . . . . . . . . . . . . . 14 (𝑥𝐴𝑥 = ((𝑧𝐴 ↦ {𝑧}) “ 𝑥))
3511unieqd 4412 . . . . . . . . . . . . . 14 (𝑥𝐴 ((𝑧𝐴 ↦ {𝑧}) “ 𝑥) = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}})
3634, 35eqtrd 2655 . . . . . . . . . . . . 13 (𝑥𝐴𝑥 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}})
3736adantl 482 . . . . . . . . . . . 12 ((𝐶𝐵𝑥𝐴) → 𝑥 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}})
3827, 37jca 554 . . . . . . . . . . 11 ((𝐶𝐵𝑥𝐴) → ({𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐵𝑥 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}}))
39 sseq1 3605 . . . . . . . . . . . 12 (𝑦 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} → (𝑦𝐵 ↔ {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} ⊆ 𝐵))
40 unieq 4410 . . . . . . . . . . . . 13 (𝑦 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} → 𝑦 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}})
4140eqeq2d 2631 . . . . . . . . . . . 12 (𝑦 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} → (𝑥 = 𝑦𝑥 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}}))
4239, 41anbi12d 746 . . . . . . . . . . 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 1843 . . . . . . . . 9 ((𝐶𝐵𝑥𝐴) → (∃𝑦 𝑦 = {𝑢 ∣ ∃𝑧𝑥 𝑢 = {𝑧}} → ∃𝑦(𝑦𝐵𝑥 = 𝑦)))
45443adant3 1079 . . . . . . . 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 1360 . . . . . 6 ((𝐶𝐵𝑥 ∈ 𝒫 𝐴𝐵 ∈ (TopOn‘𝐴)) → ∃𝑦(𝑦𝐵𝑥 = 𝑦))
48473com23 1268 . . . . 5 ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴) ∧ 𝑥 ∈ 𝒫 𝐴) → ∃𝑦(𝑦𝐵𝑥 = 𝑦))
49483expia 1264 . . . 4 ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴)) → (𝑥 ∈ 𝒫 𝐴 → ∃𝑦(𝑦𝐵𝑥 = 𝑦)))
50 topontop 20641 . . . . . . . 8 (𝐵 ∈ (TopOn‘𝐴) → 𝐵 ∈ Top)
51 tgtop 20688 . . . . . . . 8 (𝐵 ∈ Top → (topGen‘𝐵) = 𝐵)
5250, 51syl 17 . . . . . . 7 (𝐵 ∈ (TopOn‘𝐴) → (topGen‘𝐵) = 𝐵)
5352eleq2d 2684 . . . . . 6 (𝐵 ∈ (TopOn‘𝐴) → (𝑥 ∈ (topGen‘𝐵) ↔ 𝑥𝐵))
54 eltg3 20677 . . . . . 6 (𝐵 ∈ (TopOn‘𝐴) → (𝑥 ∈ (topGen‘𝐵) ↔ ∃𝑦(𝑦𝐵𝑥 = 𝑦)))
5553, 54bitr3d 270 . . . . 5 (𝐵 ∈ (TopOn‘𝐴) → (𝑥𝐵 ↔ ∃𝑦(𝑦𝐵𝑥 = 𝑦)))
5655adantl 482 . . . 4 ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴)) → (𝑥𝐵 ↔ ∃𝑦(𝑦𝐵𝑥 = 𝑦)))
5749, 56sylibrd 249 . . 3 ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴)) → (𝑥 ∈ 𝒫 𝐴𝑥𝐵))
5857ssrdv 3589 . 2 ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴)) → 𝒫 𝐴𝐵)
593, 58eqssd 3600 1 ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴)) → 𝐵 = 𝒫 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1036   = wceq 1480  ∃wex 1701   ∈ wcel 1987  {cab 2607  ∃wrex 2908  Vcvv 3186   ⊆ wss 3555  ∅c0 3891  𝒫 cpw 4130  {csn 4148  {cpr 4150  ∪ cuni 4402   ↦ cmpt 4673  ran crn 5075   ↾ cres 5076   “ cima 5077  ‘cfv 5847  topGenctg 16019  Topctop 20617  TopOnctopon 20618 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fv 5855  df-topgen 16025  df-top 20621  df-topon 20623 This theorem is referenced by:  dissneq  32820
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