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Theorem dissneqlem 36525
Description: This is the core of the proof of dissneq 36526, 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 22653 . . . 4 (𝐡 ∈ (TopOnβ€˜π΄) β†’ ({βˆ…, 𝐴} βŠ† 𝐡 ∧ 𝐡 βŠ† 𝒫 𝐴))
21adantl 481 . . 3 ((𝐢 βŠ† 𝐡 ∧ 𝐡 ∈ (TopOnβ€˜π΄)) β†’ ({βˆ…, 𝐴} βŠ† 𝐡 ∧ 𝐡 βŠ† 𝒫 𝐴))
32simprd 495 . 2 ((𝐢 βŠ† 𝐡 ∧ 𝐡 ∈ (TopOnβ€˜π΄)) β†’ 𝐡 βŠ† 𝒫 𝐴)
4 velpw 4608 . . . . . . 7 (π‘₯ ∈ 𝒫 𝐴 ↔ π‘₯ βŠ† 𝐴)
5 simp3 1137 . . . . . . . . . 10 ((𝐢 βŠ† 𝐡 ∧ π‘₯ βŠ† 𝐴 ∧ 𝐡 ∈ (TopOnβ€˜π΄)) β†’ 𝐡 ∈ (TopOnβ€˜π΄))
6 df-ima 5690 . . . . . . . . . . . . . . . . . 18 ((𝑧 ∈ 𝐴 ↦ {𝑧}) β€œ π‘₯) = ran ((𝑧 ∈ 𝐴 ↦ {𝑧}) β†Ύ π‘₯)
7 resmpt 6038 . . . . . . . . . . . . . . . . . . 19 (π‘₯ βŠ† 𝐴 β†’ ((𝑧 ∈ 𝐴 ↦ {𝑧}) β†Ύ π‘₯) = (𝑧 ∈ π‘₯ ↦ {𝑧}))
87rneqd 5938 . . . . . . . . . . . . . . . . . 18 (π‘₯ βŠ† 𝐴 β†’ ran ((𝑧 ∈ 𝐴 ↦ {𝑧}) β†Ύ π‘₯) = ran (𝑧 ∈ π‘₯ ↦ {𝑧}))
96, 8eqtrid 2783 . . . . . . . . . . . . . . . . 17 (π‘₯ βŠ† 𝐴 β†’ ((𝑧 ∈ 𝐴 ↦ {𝑧}) β€œ π‘₯) = ran (𝑧 ∈ π‘₯ ↦ {𝑧}))
10 rnmptsn 36520 . . . . . . . . . . . . . . . . 17 ran (𝑧 ∈ π‘₯ ↦ {𝑧}) = {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}}
119, 10eqtrdi 2787 . . . . . . . . . . . . . . . 16 (π‘₯ βŠ† 𝐴 β†’ ((𝑧 ∈ 𝐴 ↦ {𝑧}) β€œ π‘₯) = {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}})
12 imassrn 6071 . . . . . . . . . . . . . . . 16 ((𝑧 ∈ 𝐴 ↦ {𝑧}) β€œ π‘₯) βŠ† ran (𝑧 ∈ 𝐴 ↦ {𝑧})
1311, 12eqsstrrdi 4038 . . . . . . . . . . . . . . 15 (π‘₯ βŠ† 𝐴 β†’ {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}} βŠ† ran (𝑧 ∈ 𝐴 ↦ {𝑧}))
14 rnmptsn 36520 . . . . . . . . . . . . . . 15 ran (𝑧 ∈ 𝐴 ↦ {𝑧}) = {𝑒 ∣ βˆƒπ‘§ ∈ 𝐴 𝑒 = {𝑧}}
1513, 14sseqtrdi 4033 . . . . . . . . . . . . . 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 4639 . . . . . . . . . . . . . . . . . 18 (π‘₯ = 𝑧 β†’ {π‘₯} = {𝑧})
1817eqeq2d 2742 . . . . . . . . . . . . . . . . 17 (π‘₯ = 𝑧 β†’ (𝑒 = {π‘₯} ↔ 𝑒 = {𝑧}))
1918cbvrexvw 3234 . . . . . . . . . . . . . . . 16 (βˆƒπ‘₯ ∈ 𝐴 𝑒 = {π‘₯} ↔ βˆƒπ‘§ ∈ 𝐴 𝑒 = {𝑧})
2019abbii 2801 . . . . . . . . . . . . . . 15 {𝑒 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑒 = {π‘₯}} = {𝑒 ∣ βˆƒπ‘§ ∈ 𝐴 𝑒 = {𝑧}}
2116, 20eqtri 2759 . . . . . . . . . . . . . 14 𝐢 = {𝑒 ∣ βˆƒπ‘§ ∈ 𝐴 𝑒 = {𝑧}}
2215, 21sseqtrrdi 4034 . . . . . . . . . . . . 13 (π‘₯ βŠ† 𝐴 β†’ {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}} βŠ† 𝐢)
2322adantl 481 . . . . . . . . . . . 12 ((𝐢 βŠ† 𝐡 ∧ π‘₯ βŠ† 𝐴) β†’ {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}} βŠ† 𝐢)
24 sstr 3991 . . . . . . . . . . . . . 14 (({𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}} βŠ† 𝐢 ∧ 𝐢 βŠ† 𝐡) β†’ {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}} βŠ† 𝐡)
2524expcom 413 . . . . . . . . . . . . 13 (𝐢 βŠ† 𝐡 β†’ ({𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}} βŠ† 𝐢 β†’ {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}} βŠ† 𝐡))
2625adantr 480 . . . . . . . . . . . 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 1131 . . . . . . . . . 10 ((𝐢 βŠ† 𝐡 ∧ π‘₯ βŠ† 𝐴 ∧ 𝐡 ∈ (TopOnβ€˜π΄)) β†’ {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}} βŠ† 𝐡)
295, 28ssexd 5325 . . . . . . . . 9 ((𝐢 βŠ† 𝐡 ∧ π‘₯ βŠ† 𝐴 ∧ 𝐡 ∈ (TopOnβ€˜π΄)) β†’ {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}} ∈ V)
30 isset 3486 . . . . . . . . 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 2731 . . . . . . . . . . . . . . 15 (𝑧 ∈ 𝐴 ↦ {𝑧}) = (𝑧 ∈ 𝐴 ↦ {𝑧})
33 eqid 2731 . . . . . . . . . . . . . . 15 {𝑒 ∣ βˆƒπ‘§ ∈ 𝐴 𝑒 = {𝑧}} = {𝑒 ∣ βˆƒπ‘§ ∈ 𝐴 𝑒 = {𝑧}}
3432, 33mptsnun 36524 . . . . . . . . . . . . . 14 (π‘₯ βŠ† 𝐴 β†’ π‘₯ = βˆͺ ((𝑧 ∈ 𝐴 ↦ {𝑧}) β€œ π‘₯))
3511unieqd 4923 . . . . . . . . . . . . . 14 (π‘₯ βŠ† 𝐴 β†’ βˆͺ ((𝑧 ∈ 𝐴 ↦ {𝑧}) β€œ π‘₯) = βˆͺ {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}})
3634, 35eqtrd 2771 . . . . . . . . . . . . 13 (π‘₯ βŠ† 𝐴 β†’ π‘₯ = βˆͺ {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}})
3736adantl 481 . . . . . . . . . . . 12 ((𝐢 βŠ† 𝐡 ∧ π‘₯ βŠ† 𝐴) β†’ π‘₯ = βˆͺ {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}})
3827, 37jca 511 . . . . . . . . . . 11 ((𝐢 βŠ† 𝐡 ∧ π‘₯ βŠ† 𝐴) β†’ ({𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}} βŠ† 𝐡 ∧ π‘₯ = βˆͺ {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}}))
39 sseq1 4008 . . . . . . . . . . . 12 (𝑦 = {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}} β†’ (𝑦 βŠ† 𝐡 ↔ {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}} βŠ† 𝐡))
40 unieq 4920 . . . . . . . . . . . . 13 (𝑦 = {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}} β†’ βˆͺ 𝑦 = βˆͺ {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}})
4140eqeq2d 2742 . . . . . . . . . . . 12 (𝑦 = {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}} β†’ (π‘₯ = βˆͺ 𝑦 ↔ π‘₯ = βˆͺ {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}}))
4239, 41anbi12d 630 . . . . . . . . . . 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 1919 . . . . . . . . 9 ((𝐢 βŠ† 𝐡 ∧ π‘₯ βŠ† 𝐴) β†’ (βˆƒπ‘¦ 𝑦 = {𝑒 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑒 = {𝑧}} β†’ βˆƒπ‘¦(𝑦 βŠ† 𝐡 ∧ π‘₯ = βˆͺ 𝑦)))
45443adant3 1131 . . . . . . . 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 1403 . . . . . 6 ((𝐢 βŠ† 𝐡 ∧ π‘₯ ∈ 𝒫 𝐴 ∧ 𝐡 ∈ (TopOnβ€˜π΄)) β†’ βˆƒπ‘¦(𝑦 βŠ† 𝐡 ∧ π‘₯ = βˆͺ 𝑦))
48473com23 1125 . . . . 5 ((𝐢 βŠ† 𝐡 ∧ 𝐡 ∈ (TopOnβ€˜π΄) ∧ π‘₯ ∈ 𝒫 𝐴) β†’ βˆƒπ‘¦(𝑦 βŠ† 𝐡 ∧ π‘₯ = βˆͺ 𝑦))
49483expia 1120 . . . 4 ((𝐢 βŠ† 𝐡 ∧ 𝐡 ∈ (TopOnβ€˜π΄)) β†’ (π‘₯ ∈ 𝒫 𝐴 β†’ βˆƒπ‘¦(𝑦 βŠ† 𝐡 ∧ π‘₯ = βˆͺ 𝑦)))
50 topontop 22636 . . . . . . . 8 (𝐡 ∈ (TopOnβ€˜π΄) β†’ 𝐡 ∈ Top)
51 tgtop 22697 . . . . . . . 8 (𝐡 ∈ Top β†’ (topGenβ€˜π΅) = 𝐡)
5250, 51syl 17 . . . . . . 7 (𝐡 ∈ (TopOnβ€˜π΄) β†’ (topGenβ€˜π΅) = 𝐡)
5352eleq2d 2818 . . . . . 6 (𝐡 ∈ (TopOnβ€˜π΄) β†’ (π‘₯ ∈ (topGenβ€˜π΅) ↔ π‘₯ ∈ 𝐡))
54 eltg3 22686 . . . . . 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 481 . . . 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 3989 . 2 ((𝐢 βŠ† 𝐡 ∧ 𝐡 ∈ (TopOnβ€˜π΄)) β†’ 𝒫 𝐴 βŠ† 𝐡)
593, 58eqssd 4000 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 395   ∧ w3a 1086   = wceq 1540  βˆƒwex 1780   ∈ wcel 2105  {cab 2708  βˆƒwrex 3069  Vcvv 3473   βŠ† wss 3949  βˆ…c0 4323  π’« cpw 4603  {csn 4629  {cpr 4631  βˆͺ cuni 4909   ↦ cmpt 5232  ran crn 5678   β†Ύ cres 5679   β€œ cima 5680  β€˜cfv 6544  topGenctg 17388  Topctop 22616  TopOnctopon 22633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fv 6552  df-topgen 17394  df-top 22617  df-topon 22634
This theorem is referenced by:  dissneq  36526
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