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Theorem ssidcn 13203
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 13190 . . 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 5491 . . . . 5 ( I β†Ύ 𝑋):𝑋–1-1-onto→𝑋
3 f1of 5453 . . . . 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 303 . . 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 198 . 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 5282 . . . . . . 7 β—‘( I β†Ύ 𝑋) = ( I β†Ύ 𝑋)
87imaeq1i 4960 . . . . . 6 (β—‘( I β†Ύ 𝑋) β€œ π‘₯) = (( I β†Ύ 𝑋) β€œ π‘₯)
9 elssuni 3833 . . . . . . . . 9 (π‘₯ ∈ 𝐾 β†’ π‘₯ βŠ† βˆͺ 𝐾)
109adantl 277 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ π‘₯ ∈ 𝐾) β†’ π‘₯ βŠ† βˆͺ 𝐾)
11 toponuni 13006 . . . . . . . . 9 (𝐾 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐾)
1211ad2antlr 489 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ π‘₯ ∈ 𝐾) β†’ 𝑋 = βˆͺ 𝐾)
1310, 12sseqtrrd 3192 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ π‘₯ ∈ 𝐾) β†’ π‘₯ βŠ† 𝑋)
14 resiima 4979 . . . . . . 7 (π‘₯ βŠ† 𝑋 β†’ (( I β†Ύ 𝑋) β€œ π‘₯) = π‘₯)
1513, 14syl 14 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ π‘₯ ∈ 𝐾) β†’ (( I β†Ύ 𝑋) β€œ π‘₯) = π‘₯)
168, 15eqtrid 2220 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ π‘₯ ∈ 𝐾) β†’ (β—‘( I β†Ύ 𝑋) β€œ π‘₯) = π‘₯)
1716eleq1d 2244 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) ∧ π‘₯ ∈ 𝐾) β†’ ((β—‘( I β†Ύ 𝑋) β€œ π‘₯) ∈ 𝐽 ↔ π‘₯ ∈ 𝐽))
1817ralbidva 2471 . . 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 3143 . . 3 (𝐾 βŠ† 𝐽 ↔ βˆ€π‘₯ ∈ 𝐾 π‘₯ ∈ 𝐽)
2018, 19bitr4di 198 . 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 188 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹)) β†’ (( I β†Ύ 𝑋) ∈ (𝐽 Cn 𝐾) ↔ 𝐾 βŠ† 𝐽))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2146  βˆ€wral 2453   βŠ† wss 3127  βˆͺ cuni 3805   I cid 4282  β—‘ccnv 4619   β†Ύ cres 4622   β€œ cima 4623  βŸΆwf 5204  β€“1-1-ontoβ†’wf1o 5207  β€˜cfv 5208  (class class class)co 5865  TopOnctopon 13001   Cn ccn 13178
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-map 6640  df-top 12989  df-topon 13002  df-cn 13181
This theorem is referenced by:  idcn  13205
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