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Theorem ist0-4 23734
Description: The topological indistinguishability map is injective iff the space is T0. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
kqval.2 𝐹 = (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦})
Assertion
Ref Expression
ist0-4 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (𝐽 ∈ Kol2 ↔ 𝐹:𝑋–1-1β†’V))
Distinct variable groups:   π‘₯,𝑦,𝐽   π‘₯,𝑋,𝑦
Allowed substitution hints:   𝐹(π‘₯,𝑦)

Proof of Theorem ist0-4
Dummy variables 𝑀 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 kqval.2 . . . . . 6 𝐹 = (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦})
21kqfeq 23729 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑧 ∈ 𝑋 ∧ 𝑀 ∈ 𝑋) β†’ ((πΉβ€˜π‘§) = (πΉβ€˜π‘€) ↔ βˆ€π‘¦ ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝑀 ∈ 𝑦)))
323expb 1117 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (𝑧 ∈ 𝑋 ∧ 𝑀 ∈ 𝑋)) β†’ ((πΉβ€˜π‘§) = (πΉβ€˜π‘€) ↔ βˆ€π‘¦ ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝑀 ∈ 𝑦)))
43imbi1d 340 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (𝑧 ∈ 𝑋 ∧ 𝑀 ∈ 𝑋)) β†’ (((πΉβ€˜π‘§) = (πΉβ€˜π‘€) β†’ 𝑧 = 𝑀) ↔ (βˆ€π‘¦ ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝑀 ∈ 𝑦) β†’ 𝑧 = 𝑀)))
542ralbidva 3210 . 2 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (βˆ€π‘§ ∈ 𝑋 βˆ€π‘€ ∈ 𝑋 ((πΉβ€˜π‘§) = (πΉβ€˜π‘€) β†’ 𝑧 = 𝑀) ↔ βˆ€π‘§ ∈ 𝑋 βˆ€π‘€ ∈ 𝑋 (βˆ€π‘¦ ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝑀 ∈ 𝑦) β†’ 𝑧 = 𝑀)))
61kqffn 23730 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐹 Fn 𝑋)
7 dffn2 6731 . . . 4 (𝐹 Fn 𝑋 ↔ 𝐹:π‘‹βŸΆV)
86, 7sylib 217 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐹:π‘‹βŸΆV)
9 dff13 7271 . . . 4 (𝐹:𝑋–1-1β†’V ↔ (𝐹:π‘‹βŸΆV ∧ βˆ€π‘§ ∈ 𝑋 βˆ€π‘€ ∈ 𝑋 ((πΉβ€˜π‘§) = (πΉβ€˜π‘€) β†’ 𝑧 = 𝑀)))
109baib 534 . . 3 (𝐹:π‘‹βŸΆV β†’ (𝐹:𝑋–1-1β†’V ↔ βˆ€π‘§ ∈ 𝑋 βˆ€π‘€ ∈ 𝑋 ((πΉβ€˜π‘§) = (πΉβ€˜π‘€) β†’ 𝑧 = 𝑀)))
118, 10syl 17 . 2 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (𝐹:𝑋–1-1β†’V ↔ βˆ€π‘§ ∈ 𝑋 βˆ€π‘€ ∈ 𝑋 ((πΉβ€˜π‘§) = (πΉβ€˜π‘€) β†’ 𝑧 = 𝑀)))
12 ist0-2 23349 . 2 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (𝐽 ∈ Kol2 ↔ βˆ€π‘§ ∈ 𝑋 βˆ€π‘€ ∈ 𝑋 (βˆ€π‘¦ ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝑀 ∈ 𝑦) β†’ 𝑧 = 𝑀)))
135, 11, 123bitr4rd 311 1 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (𝐽 ∈ Kol2 ↔ 𝐹:𝑋–1-1β†’V))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1534   ∈ wcel 2100  βˆ€wral 3054  {crab 3427  Vcvv 3471   ↦ cmpt 5236   Fn wfn 6550  βŸΆwf 6551  β€“1-1β†’wf1 6552  β€˜cfv 6555  TopOnctopon 22913  Kol2ct0 23311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2102  ax-9 2110  ax-10 2133  ax-11 2150  ax-12 2170  ax-ext 2700  ax-sep 5304  ax-nul 5311  ax-pow 5370  ax-pr 5434  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2062  df-mo 2532  df-eu 2561  df-clab 2707  df-cleq 2721  df-clel 2806  df-nfc 2881  df-ne 2934  df-ral 3055  df-rex 3064  df-rab 3428  df-v 3473  df-dif 3959  df-un 3961  df-in 3963  df-ss 3973  df-nul 4333  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4916  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5581  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-iota 6507  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fv 6563  df-topon 22914  df-t0 23318
This theorem is referenced by:  t0kq  23823
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