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Theorem idqtop 23210
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 6628 . . . . . . 7 β—‘( I β†Ύ 𝑋) = ( I β†Ύ 𝑋)
21imaeq1i 6057 . . . . . 6 (β—‘( I β†Ύ 𝑋) β€œ π‘₯) = (( I β†Ύ 𝑋) β€œ π‘₯)
3 resiima 6076 . . . . . . 7 (π‘₯ βŠ† 𝑋 β†’ (( I β†Ύ 𝑋) β€œ π‘₯) = π‘₯)
43adantl 483 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘₯ βŠ† 𝑋) β†’ (( I β†Ύ 𝑋) β€œ π‘₯) = π‘₯)
52, 4eqtrid 2785 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘₯ βŠ† 𝑋) β†’ (β—‘( I β†Ύ 𝑋) β€œ π‘₯) = π‘₯)
65eleq1d 2819 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘₯ βŠ† 𝑋) β†’ ((β—‘( I β†Ύ 𝑋) β€œ π‘₯) ∈ 𝐽 ↔ π‘₯ ∈ 𝐽))
76pm5.32da 580 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ ((π‘₯ βŠ† 𝑋 ∧ (β—‘( I β†Ύ 𝑋) β€œ π‘₯) ∈ 𝐽) ↔ (π‘₯ βŠ† 𝑋 ∧ π‘₯ ∈ 𝐽)))
8 f1oi 6872 . . . . 5 ( I β†Ύ 𝑋):𝑋–1-1-onto→𝑋
9 f1ofo 6841 . . . . 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 23207 . . . 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 22429 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘₯ ∈ 𝐽) β†’ π‘₯ βŠ† 𝑋)
1413ex 414 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (π‘₯ ∈ 𝐽 β†’ π‘₯ βŠ† 𝑋))
1514pm4.71rd 564 . . 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 2731 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 397   = wceq 1542   ∈ wcel 2107   βŠ† wss 3949   I cid 5574  β—‘ccnv 5676   β†Ύ cres 5679   β€œ cima 5680  β€“ontoβ†’wfo 6542  β€“1-1-ontoβ†’wf1o 6543  β€˜cfv 6544  (class class class)co 7409   qTop cqtop 17449  TopOnctopon 22412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  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-iun 5000  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-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-qtop 17453  df-topon 22413
This theorem is referenced by: (None)
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