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Theorem pi1val 19054
 Description: The definition of the fundamental group. (Contributed by Mario Carneiro, 11-Feb-2015.) (Revised by Mario Carneiro, 10-Jul-2015.)
Hypotheses
Ref Expression
pi1val.g
pi1val.1 TopOn
pi1val.2
pi1val.o
Assertion
Ref Expression
pi1val s

Proof of Theorem pi1val
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pi1val.g . 2
2 df-pi1 19025 . . . 4 s
32a1i 11 . . 3 s
4 simprl 733 . . . . . 6
5 simprr 734 . . . . . 6
64, 5oveq12d 6091 . . . . 5
7 pi1val.o . . . . 5
86, 7syl6eqr 2485 . . . 4
94fveq2d 5724 . . . 4
108, 9oveq12d 6091 . . 3 s s
11 unieq 4016 . . . . 5
1211adantl 453 . . . 4
13 pi1val.1 . . . . . 6 TopOn
14 toponuni 16984 . . . . . 6 TopOn
1513, 14syl 16 . . . . 5
1615adantr 452 . . . 4
1712, 16eqtr4d 2470 . . 3
18 topontop 16983 . . . 4 TopOn
1913, 18syl 16 . . 3
20 pi1val.2 . . 3
21 ovex 6098 . . . 4 s
2221a1i 11 . . 3 s
233, 10, 17, 19, 20, 22ovmpt2dx 6192 . 2 s
241, 23syl5eq 2479 1 s
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1652   wcel 1725  cvv 2948  cuni 4007  cfv 5446  (class class class)co 6073   cmpt2 6075   s cqus 13723  ctop 16950  TopOnctopon 16951   cphtpc 18986   comi 19018   cpi1 19020 This theorem is referenced by:  pi1bas  19055  pi1addf  19064  pi1addval  19065  pi1grplem  19066 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-topon 16958  df-pi1 19025
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