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Theorem idqtop 23561
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 6620 . . . . . . 7 β—‘( I β†Ύ 𝑋) = ( I β†Ύ 𝑋)
21imaeq1i 6049 . . . . . 6 (β—‘( I β†Ύ 𝑋) β€œ π‘₯) = (( I β†Ύ 𝑋) β€œ π‘₯)
3 resiima 6068 . . . . . . 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 2778 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘₯ βŠ† 𝑋) β†’ (β—‘( I β†Ύ 𝑋) β€œ π‘₯) = π‘₯)
65eleq1d 2812 . . . 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 6864 . . . . 5 ( I β†Ύ 𝑋):𝑋–1-1-onto→𝑋
9 f1ofo 6833 . . . . 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 23558 . . . 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 684 . . 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 22780 . . . . 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 2724 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 1533   ∈ wcel 2098   βŠ† wss 3943   I cid 5566  β—‘ccnv 5668   β†Ύ cres 5671   β€œ cima 5672  β€“ontoβ†’wfo 6534  β€“1-1-ontoβ†’wf1o 6535  β€˜cfv 6536  (class class class)co 7404   qTop cqtop 17456  TopOnctopon 22763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-qtop 17460  df-topon 22764
This theorem is referenced by: (None)
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