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Theorem ssidcn 23079
Description: The identity function is a continuous function from one topology to another topology on the same set iff the domain is finer than the codomain. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
Assertion
Ref Expression
ssidcn ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) β†’ (( I β†Ύ 𝑋) ∈ (𝐽 Cn 𝐾) ↔ 𝐾 βŠ† 𝐽))
Proof of Theorem ssidcn
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 iscn 23059 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) β†’ (( I β†Ύ 𝑋) ∈ (𝐽 Cn 𝐾) ↔ (( I β†Ύ 𝑋):π‘‹βŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝐾 (β—‘( I β†Ύ 𝑋) β€œ π‘₯) ∈ 𝐽)))
2 f1oi 6871 . . . . 5 ( I β†Ύ 𝑋):𝑋–1-1-onto→𝑋
3 f1of 6833 . . . . 5 (( I β†Ύ 𝑋):𝑋–1-1-onto→𝑋 β†’ ( I β†Ύ 𝑋):π‘‹βŸΆπ‘‹)
42, 3ax-mp 5 . . . 4 ( I β†Ύ 𝑋):π‘‹βŸΆπ‘‹
54biantrur 530 . . 3 (βˆ€π‘₯ ∈ 𝐾 (β—‘( I β†Ύ 𝑋) β€œ π‘₯) ∈ 𝐽 ↔ (( I β†Ύ 𝑋):π‘‹βŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝐾 (β—‘( I β†Ύ 𝑋) β€œ π‘₯) ∈ 𝐽))
61, 5bitr4di 289 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) β†’ (( I β†Ύ 𝑋) ∈ (𝐽 Cn 𝐾) ↔ βˆ€π‘₯ ∈ 𝐾 (β—‘( I β†Ύ 𝑋) β€œ π‘₯) ∈ 𝐽))
7 cnvresid 6627 . . . . . . 7 β—‘( I β†Ύ 𝑋) = ( I β†Ύ 𝑋)
87imaeq1i 6056 . . . . . 6 (β—‘( I β†Ύ 𝑋) β€œ π‘₯) = (( I β†Ύ 𝑋) β€œ π‘₯)
9 elssuni 4941 . . . . . . . . 9 (π‘₯ ∈ 𝐾 β†’ π‘₯ βŠ† βˆͺ 𝐾)
109adantl 481 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ π‘₯ ∈ 𝐾) β†’ π‘₯ βŠ† βˆͺ 𝐾)
11 toponuni 22736 . . . . . . . . 9 (𝐾 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐾)
1211ad2antlr 724 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ π‘₯ ∈ 𝐾) β†’ 𝑋 = βˆͺ 𝐾)
1310, 12sseqtrrd 4023 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ π‘₯ ∈ 𝐾) β†’ π‘₯ βŠ† 𝑋)
14 resiima 6075 . . . . . . 7 (π‘₯ βŠ† 𝑋 β†’ (( I β†Ύ 𝑋) β€œ π‘₯) = π‘₯)
1513, 14syl 17 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ π‘₯ ∈ 𝐾) β†’ (( I β†Ύ 𝑋) β€œ π‘₯) = π‘₯)
168, 15eqtrid 2783 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ π‘₯ ∈ 𝐾) β†’ (β—‘( I β†Ύ 𝑋) β€œ π‘₯) = π‘₯)
1716eleq1d 2817 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ π‘₯ ∈ 𝐾) β†’ ((β—‘( I β†Ύ 𝑋) β€œ π‘₯) ∈ 𝐽 ↔ π‘₯ ∈ 𝐽))
1817ralbidva 3174 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) β†’ (βˆ€π‘₯ ∈ 𝐾 (β—‘( I β†Ύ 𝑋) β€œ π‘₯) ∈ 𝐽 ↔ βˆ€π‘₯ ∈ 𝐾 π‘₯ ∈ 𝐽))
19 dfss3 3970 . . 3 (𝐾 βŠ† 𝐽 ↔ βˆ€π‘₯ ∈ 𝐾 π‘₯ ∈ 𝐽)
2018, 19bitr4di 289 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) β†’ (βˆ€π‘₯ ∈ 𝐾 (β—‘( I β†Ύ 𝑋) β€œ π‘₯) ∈ 𝐽 ↔ 𝐾 βŠ† 𝐽))
216, 20bitrd 279 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) β†’ (( I β†Ύ 𝑋) ∈ (𝐽 Cn 𝐾) ↔ 𝐾 βŠ† 𝐽))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105  βˆ€wral 3060   βŠ† wss 3948  βˆͺ cuni 4908   I cid 5573  β—‘ccnv 5675   β†Ύ cres 5678   β€œ cima 5679  βŸΆwf 6539  β€“1-1-ontoβ†’wf1o 6542  β€˜cfv 6543  (class class class)co 7412  TopOnctopon 22732   Cn ccn 23048
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 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
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-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-map 8828  df-top 22716  df-topon 22733  df-cn 23051
This theorem is referenced by:  idcn  23081  sshauslem  23196
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