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Theorem dissneqlem 36524
Description: This is the core of the proof of dissneq 36525, 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 𝐢 = {𝑒 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑒 = {π‘₯}}
Assertion
Ref Expression
dissneqlem ((𝐢 βŠ† 𝐡 ∧ 𝐡 ∈ (TopOnβ€˜π΄)) β†’ 𝐡 = 𝒫 𝐴)
Distinct variable groups:   𝑒,𝐴,π‘₯   π‘₯,𝐡   π‘₯,𝐢
Allowed substitution hints:   𝐡(𝑒)   𝐢(𝑒)
Proof of Theorem dissneqlem
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 topgele 22652 . . . 4 (𝐡 ∈ (TopOnβ€˜π΄) β†’ ({βˆ…, 𝐴} βŠ† 𝐡 ∧ 𝐡 βŠ† 𝒫 𝐴))
21adantl 480 . . 3 ((𝐢 βŠ† 𝐡 ∧ 𝐡 ∈ (TopOnβ€˜π΄)) β†’ ({βˆ…, 𝐴} βŠ† 𝐡 ∧ 𝐡 βŠ† 𝒫 𝐴))
32simprd 494 . 2 ((𝐢 βŠ† 𝐡 ∧ 𝐡 ∈ (TopOnβ€˜π΄)) β†’ 𝐡 βŠ† 𝒫 𝐴)
4 velpw 4606 . . . . . . 7 (π‘₯ ∈ 𝒫 𝐴 ↔ π‘₯ βŠ† 𝐴)
5 simp3 1136 . . . . . . . . . 10 ((𝐢 βŠ† 𝐡 ∧ π‘₯ βŠ† 𝐴 ∧ 𝐡 ∈ (TopOnβ€˜π΄)) β†’ 𝐡 ∈ (TopOnβ€˜π΄))
6 df-ima 5688 . . . . . . . . . . . . . . . . . 18 ((𝑧 ∈ 𝐴 ↦ {𝑧}) β€œ π‘₯) = ran ((𝑧 ∈ 𝐴 ↦ {𝑧}) β†Ύ π‘₯)
7 resmpt 6036 . . . . . . . . . . . . . . . . . . 19 (π‘₯ βŠ† 𝐴 β†’ ((𝑧 ∈ 𝐴 ↦ {𝑧}) β†Ύ π‘₯) = (𝑧 ∈ π‘₯ ↦ {𝑧}))
87rneqd 5936 . . . . . . . . . . . . . . . . . 18 (π‘₯ βŠ† 𝐴 β†’ ran ((𝑧 ∈ 𝐴 ↦ {𝑧}) β†Ύ π‘₯) = ran (𝑧 ∈ π‘₯ ↦ {𝑧}))
96, 8eqtrid 2782 . . . . . . . . . . . . . . . . 17 (π‘₯ βŠ† 𝐴 β†’ ((𝑧 ∈ 𝐴 ↦ {𝑧}) β€œ π‘₯) = ran (𝑧 ∈ π‘₯ ↦ {𝑧}))
10 rnmptsn 36519 . . . . . . . . . . . . . . . . 17 ran (𝑧 ∈ π‘₯ ↦ {𝑧}) = {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}}
119, 10eqtrdi 2786 . . . . . . . . . . . . . . . 16 (π‘₯ βŠ† 𝐴 β†’ ((𝑧 ∈ 𝐴 ↦ {𝑧}) β€œ π‘₯) = {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}})
12 imassrn 6069 . . . . . . . . . . . . . . . 16 ((𝑧 ∈ 𝐴 ↦ {𝑧}) β€œ π‘₯) βŠ† ran (𝑧 ∈ 𝐴 ↦ {𝑧})
1311, 12eqsstrrdi 4036 . . . . . . . . . . . . . . 15 (π‘₯ βŠ† 𝐴 β†’ {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}} βŠ† ran (𝑧 ∈ 𝐴 ↦ {𝑧}))
14 rnmptsn 36519 . . . . . . . . . . . . . . 15 ran (𝑧 ∈ 𝐴 ↦ {𝑧}) = {𝑒 ∣ βˆƒπ‘§ ∈ 𝐴 𝑒 = {𝑧}}
1513, 14sseqtrdi 4031 . . . . . . . . . . . . . 14 (π‘₯ βŠ† 𝐴 β†’ {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}} βŠ† {𝑒 ∣ βˆƒπ‘§ ∈ 𝐴 𝑒 = {𝑧}})
16 dissneq.c . . . . . . . . . . . . . . 15 𝐢 = {𝑒 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑒 = {π‘₯}}
17 sneq 4637 . . . . . . . . . . . . . . . . . 18 (π‘₯ = 𝑧 β†’ {π‘₯} = {𝑧})
1817eqeq2d 2741 . . . . . . . . . . . . . . . . 17 (π‘₯ = 𝑧 β†’ (𝑒 = {π‘₯} ↔ 𝑒 = {𝑧}))
1918cbvrexvw 3233 . . . . . . . . . . . . . . . 16 (βˆƒπ‘₯ ∈ 𝐴 𝑒 = {π‘₯} ↔ βˆƒπ‘§ ∈ 𝐴 𝑒 = {𝑧})
2019abbii 2800 . . . . . . . . . . . . . . 15 {𝑒 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑒 = {π‘₯}} = {𝑒 ∣ βˆƒπ‘§ ∈ 𝐴 𝑒 = {𝑧}}
2116, 20eqtri 2758 . . . . . . . . . . . . . 14 𝐢 = {𝑒 ∣ βˆƒπ‘§ ∈ 𝐴 𝑒 = {𝑧}}
2215, 21sseqtrrdi 4032 . . . . . . . . . . . . 13 (π‘₯ βŠ† 𝐴 β†’ {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}} βŠ† 𝐢)
2322adantl 480 . . . . . . . . . . . 12 ((𝐢 βŠ† 𝐡 ∧ π‘₯ βŠ† 𝐴) β†’ {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}} βŠ† 𝐢)
24 sstr 3989 . . . . . . . . . . . . . 14 (({𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}} βŠ† 𝐢 ∧ 𝐢 βŠ† 𝐡) β†’ {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}} βŠ† 𝐡)
2524expcom 412 . . . . . . . . . . . . 13 (𝐢 βŠ† 𝐡 β†’ ({𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}} βŠ† 𝐢 β†’ {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}} βŠ† 𝐡))
2625adantr 479 . . . . . . . . . . . 12 ((𝐢 βŠ† 𝐡 ∧ π‘₯ βŠ† 𝐴) β†’ ({𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}} βŠ† 𝐢 β†’ {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}} βŠ† 𝐡))
2723, 26mpd 15 . . . . . . . . . . 11 ((𝐢 βŠ† 𝐡 ∧ π‘₯ βŠ† 𝐴) β†’ {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}} βŠ† 𝐡)
28273adant3 1130 . . . . . . . . . 10 ((𝐢 βŠ† 𝐡 ∧ π‘₯ βŠ† 𝐴 ∧ 𝐡 ∈ (TopOnβ€˜π΄)) β†’ {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}} βŠ† 𝐡)
295, 28ssexd 5323 . . . . . . . . 9 ((𝐢 βŠ† 𝐡 ∧ π‘₯ βŠ† 𝐴 ∧ 𝐡 ∈ (TopOnβ€˜π΄)) β†’ {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}} ∈ V)
30 isset 3485 . . . . . . . . 9 ({𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}} ∈ V ↔ βˆƒπ‘¦ 𝑦 = {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}})
3129, 30sylib 217 . . . . . . . 8 ((𝐢 βŠ† 𝐡 ∧ π‘₯ βŠ† 𝐴 ∧ 𝐡 ∈ (TopOnβ€˜π΄)) β†’ βˆƒπ‘¦ 𝑦 = {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}})
32 eqid 2730 . . . . . . . . . . . . . . 15 (𝑧 ∈ 𝐴 ↦ {𝑧}) = (𝑧 ∈ 𝐴 ↦ {𝑧})
33 eqid 2730 . . . . . . . . . . . . . . 15 {𝑒 ∣ βˆƒπ‘§ ∈ 𝐴 𝑒 = {𝑧}} = {𝑒 ∣ βˆƒπ‘§ ∈ 𝐴 𝑒 = {𝑧}}
3432, 33mptsnun 36523 . . . . . . . . . . . . . 14 (π‘₯ βŠ† 𝐴 β†’ π‘₯ = βˆͺ ((𝑧 ∈ 𝐴 ↦ {𝑧}) β€œ π‘₯))
3511unieqd 4921 . . . . . . . . . . . . . 14 (π‘₯ βŠ† 𝐴 β†’ βˆͺ ((𝑧 ∈ 𝐴 ↦ {𝑧}) β€œ π‘₯) = βˆͺ {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}})
3634, 35eqtrd 2770 . . . . . . . . . . . . 13 (π‘₯ βŠ† 𝐴 β†’ π‘₯ = βˆͺ {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}})
3736adantl 480 . . . . . . . . . . . 12 ((𝐢 βŠ† 𝐡 ∧ π‘₯ βŠ† 𝐴) β†’ π‘₯ = βˆͺ {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}})
3827, 37jca 510 . . . . . . . . . . 11 ((𝐢 βŠ† 𝐡 ∧ π‘₯ βŠ† 𝐴) β†’ ({𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}} βŠ† 𝐡 ∧ π‘₯ = βˆͺ {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}}))
39 sseq1 4006 . . . . . . . . . . . 12 (𝑦 = {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}} β†’ (𝑦 βŠ† 𝐡 ↔ {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}} βŠ† 𝐡))
40 unieq 4918 . . . . . . . . . . . . 13 (𝑦 = {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}} β†’ βˆͺ 𝑦 = βˆͺ {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}})
4140eqeq2d 2741 . . . . . . . . . . . 12 (𝑦 = {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}} β†’ (π‘₯ = βˆͺ 𝑦 ↔ π‘₯ = βˆͺ {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}}))
4239, 41anbi12d 629 . . . . . . . . . . 11 (𝑦 = {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}} β†’ ((𝑦 βŠ† 𝐡 ∧ π‘₯ = βˆͺ 𝑦) ↔ ({𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}} βŠ† 𝐡 ∧ π‘₯ = βˆͺ {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}})))
4338, 42syl5ibrcom 246 . . . . . . . . . 10 ((𝐢 βŠ† 𝐡 ∧ π‘₯ βŠ† 𝐴) β†’ (𝑦 = {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}} β†’ (𝑦 βŠ† 𝐡 ∧ π‘₯ = βˆͺ 𝑦)))
4443eximdv 1918 . . . . . . . . 9 ((𝐢 βŠ† 𝐡 ∧ π‘₯ βŠ† 𝐴) β†’ (βˆƒπ‘¦ 𝑦 = {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}} β†’ βˆƒπ‘¦(𝑦 βŠ† 𝐡 ∧ π‘₯ = βˆͺ 𝑦)))
45443adant3 1130 . . . . . . . 8 ((𝐢 βŠ† 𝐡 ∧ π‘₯ βŠ† 𝐴 ∧ 𝐡 ∈ (TopOnβ€˜π΄)) β†’ (βˆƒπ‘¦ 𝑦 = {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}} β†’ βˆƒπ‘¦(𝑦 βŠ† 𝐡 ∧ π‘₯ = βˆͺ 𝑦)))
4631, 45mpd 15 . . . . . . 7 ((𝐢 βŠ† 𝐡 ∧ π‘₯ βŠ† 𝐴 ∧ 𝐡 ∈ (TopOnβ€˜π΄)) β†’ βˆƒπ‘¦(𝑦 βŠ† 𝐡 ∧ π‘₯ = βˆͺ 𝑦))
474, 46syl3an2b 1402 . . . . . 6 ((𝐢 βŠ† 𝐡 ∧ π‘₯ ∈ 𝒫 𝐴 ∧ 𝐡 ∈ (TopOnβ€˜π΄)) β†’ βˆƒπ‘¦(𝑦 βŠ† 𝐡 ∧ π‘₯ = βˆͺ 𝑦))
48473com23 1124 . . . . 5 ((𝐢 βŠ† 𝐡 ∧ 𝐡 ∈ (TopOnβ€˜π΄) ∧ π‘₯ ∈ 𝒫 𝐴) β†’ βˆƒπ‘¦(𝑦 βŠ† 𝐡 ∧ π‘₯ = βˆͺ 𝑦))
49483expia 1119 . . . 4 ((𝐢 βŠ† 𝐡 ∧ 𝐡 ∈ (TopOnβ€˜π΄)) β†’ (π‘₯ ∈ 𝒫 𝐴 β†’ βˆƒπ‘¦(𝑦 βŠ† 𝐡 ∧ π‘₯ = βˆͺ 𝑦)))
50 topontop 22635 . . . . . . . 8 (𝐡 ∈ (TopOnβ€˜π΄) β†’ 𝐡 ∈ Top)
51 tgtop 22696 . . . . . . . 8 (𝐡 ∈ Top β†’ (topGenβ€˜π΅) = 𝐡)
5250, 51syl 17 . . . . . . 7 (𝐡 ∈ (TopOnβ€˜π΄) β†’ (topGenβ€˜π΅) = 𝐡)
5352eleq2d 2817 . . . . . 6 (𝐡 ∈ (TopOnβ€˜π΄) β†’ (π‘₯ ∈ (topGenβ€˜π΅) ↔ π‘₯ ∈ 𝐡))
54 eltg3 22685 . . . . . 6 (𝐡 ∈ (TopOnβ€˜π΄) β†’ (π‘₯ ∈ (topGenβ€˜π΅) ↔ βˆƒπ‘¦(𝑦 βŠ† 𝐡 ∧ π‘₯ = βˆͺ 𝑦)))
5553, 54bitr3d 280 . . . . 5 (𝐡 ∈ (TopOnβ€˜π΄) β†’ (π‘₯ ∈ 𝐡 ↔ βˆƒπ‘¦(𝑦 βŠ† 𝐡 ∧ π‘₯ = βˆͺ 𝑦)))
5655adantl 480 . . . 4 ((𝐢 βŠ† 𝐡 ∧ 𝐡 ∈ (TopOnβ€˜π΄)) β†’ (π‘₯ ∈ 𝐡 ↔ βˆƒπ‘¦(𝑦 βŠ† 𝐡 ∧ π‘₯ = βˆͺ 𝑦)))
5749, 56sylibrd 258 . . 3 ((𝐢 βŠ† 𝐡 ∧ 𝐡 ∈ (TopOnβ€˜π΄)) β†’ (π‘₯ ∈ 𝒫 𝐴 β†’ π‘₯ ∈ 𝐡))
5857ssrdv 3987 . 2 ((𝐢 βŠ† 𝐡 ∧ 𝐡 ∈ (TopOnβ€˜π΄)) β†’ 𝒫 𝐴 βŠ† 𝐡)
593, 58eqssd 3998 1 ((𝐢 βŠ† 𝐡 ∧ 𝐡 ∈ (TopOnβ€˜π΄)) β†’ 𝐡 = 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   = wceq 1539  βˆƒwex 1779   ∈ wcel 2104  {cab 2707  βˆƒwrex 3068  Vcvv 3472   βŠ† wss 3947  βˆ…c0 4321  π’« cpw 4601  {csn 4627  {cpr 4629  βˆͺ cuni 4907   ↦ cmpt 5230  ran crn 5676   β†Ύ cres 5677   β€œ cima 5678  β€˜cfv 6542  topGenctg 17387  Topctop 22615  TopOnctopon 22632
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fv 6550  df-topgen 17393  df-top 22616  df-topon 22633
This theorem is referenced by:  dissneq  36525
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