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Theorem idqtop 23623
Description: The quotient topology induced by the identity function is the original topology. (Contributed by Mario Carneiro, 30-Aug-2015.)
Assertion
Ref Expression
idqtop (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (𝐽 qTop ( I β†Ύ 𝑋)) = 𝐽)
Proof of Theorem idqtop
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 cnvresid 6632 . . . . . . 7 β—‘( I β†Ύ 𝑋) = ( I β†Ύ 𝑋)
21imaeq1i 6060 . . . . . 6 (β—‘( I β†Ύ 𝑋) β€œ π‘₯) = (( I β†Ύ 𝑋) β€œ π‘₯)
3 resiima 6079 . . . . . . 7 (π‘₯ βŠ† 𝑋 β†’ (( I β†Ύ 𝑋) β€œ π‘₯) = π‘₯)
43adantl 481 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘₯ βŠ† 𝑋) β†’ (( I β†Ύ 𝑋) β€œ π‘₯) = π‘₯)
52, 4eqtrid 2780 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘₯ βŠ† 𝑋) β†’ (β—‘( I β†Ύ 𝑋) β€œ π‘₯) = π‘₯)
65eleq1d 2814 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘₯ βŠ† 𝑋) β†’ ((β—‘( I β†Ύ 𝑋) β€œ π‘₯) ∈ 𝐽 ↔ π‘₯ ∈ 𝐽))
76pm5.32da 578 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ ((π‘₯ βŠ† 𝑋 ∧ (β—‘( I β†Ύ 𝑋) β€œ π‘₯) ∈ 𝐽) ↔ (π‘₯ βŠ† 𝑋 ∧ π‘₯ ∈ 𝐽)))
8 f1oi 6877 . . . . 5 ( I β†Ύ 𝑋):𝑋–1-1-onto→𝑋
9 f1ofo 6846 . . . . 5 (( I β†Ύ 𝑋):𝑋–1-1-onto→𝑋 β†’ ( I β†Ύ 𝑋):𝑋–onto→𝑋)
108, 9mp1i 13 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ ( I β†Ύ 𝑋):𝑋–onto→𝑋)
11 elqtop3 23620 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ ( I β†Ύ 𝑋):𝑋–onto→𝑋) β†’ (π‘₯ ∈ (𝐽 qTop ( I β†Ύ 𝑋)) ↔ (π‘₯ βŠ† 𝑋 ∧ (β—‘( I β†Ύ 𝑋) β€œ π‘₯) ∈ 𝐽)))
1210, 11mpdan 686 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (π‘₯ ∈ (𝐽 qTop ( I β†Ύ 𝑋)) ↔ (π‘₯ βŠ† 𝑋 ∧ (β—‘( I β†Ύ 𝑋) β€œ π‘₯) ∈ 𝐽)))
13 toponss 22842 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘₯ ∈ 𝐽) β†’ π‘₯ βŠ† 𝑋)
1413ex 412 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (π‘₯ ∈ 𝐽 β†’ π‘₯ βŠ† 𝑋))
1514pm4.71rd 562 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (π‘₯ ∈ 𝐽 ↔ (π‘₯ βŠ† 𝑋 ∧ π‘₯ ∈ 𝐽)))
167, 12, 153bitr4d 311 . 2 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (π‘₯ ∈ (𝐽 qTop ( I β†Ύ 𝑋)) ↔ π‘₯ ∈ 𝐽))
1716eqrdv 2726 1 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (𝐽 qTop ( I β†Ύ 𝑋)) = 𝐽)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099   βŠ† wss 3947   I cid 5575  β—‘ccnv 5677   β†Ύ cres 5680   β€œ cima 5681  β€“ontoβ†’wfo 6546  β€“1-1-ontoβ†’wf1o 6547  β€˜cfv 6548  (class class class)co 7420   qTop cqtop 17485  TopOnctopon 22825
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 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-qtop 17489  df-topon 22826
This theorem is referenced by: (None)
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