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Theorem pi1val 23568
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 𝐺 = (𝐽 π1 𝑌)
pi1val.1 (𝜑𝐽 ∈ (TopOn‘𝑋))
pi1val.2 (𝜑𝑌𝑋)
pi1val.o 𝑂 = (𝐽 Ω1 𝑌)
Assertion
Ref Expression
pi1val (𝜑𝐺 = (𝑂 /s ( ≃ph𝐽)))

Proof of Theorem pi1val
Dummy variables 𝑗 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pi1val.g . 2 𝐺 = (𝐽 π1 𝑌)
2 df-pi1 23539 . . . 4 π1 = (𝑗 ∈ Top, 𝑦 𝑗 ↦ ((𝑗 Ω1 𝑦) /s ( ≃ph𝑗)))
32a1i 11 . . 3 (𝜑 → π1 = (𝑗 ∈ Top, 𝑦 𝑗 ↦ ((𝑗 Ω1 𝑦) /s ( ≃ph𝑗))))
4 simprl 767 . . . . . 6 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → 𝑗 = 𝐽)
5 simprr 769 . . . . . 6 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → 𝑦 = 𝑌)
64, 5oveq12d 7163 . . . . 5 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → (𝑗 Ω1 𝑦) = (𝐽 Ω1 𝑌))
7 pi1val.o . . . . 5 𝑂 = (𝐽 Ω1 𝑌)
86, 7syl6eqr 2871 . . . 4 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → (𝑗 Ω1 𝑦) = 𝑂)
94fveq2d 6667 . . . 4 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → ( ≃ph𝑗) = ( ≃ph𝐽))
108, 9oveq12d 7163 . . 3 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → ((𝑗 Ω1 𝑦) /s ( ≃ph𝑗)) = (𝑂 /s ( ≃ph𝐽)))
11 unieq 4838 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝐽)
1211adantl 482 . . . 4 ((𝜑𝑗 = 𝐽) → 𝑗 = 𝐽)
13 pi1val.1 . . . . . 6 (𝜑𝐽 ∈ (TopOn‘𝑋))
14 toponuni 21450 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
1513, 14syl 17 . . . . 5 (𝜑𝑋 = 𝐽)
1615adantr 481 . . . 4 ((𝜑𝑗 = 𝐽) → 𝑋 = 𝐽)
1712, 16eqtr4d 2856 . . 3 ((𝜑𝑗 = 𝐽) → 𝑗 = 𝑋)
18 topontop 21449 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
1913, 18syl 17 . . 3 (𝜑𝐽 ∈ Top)
20 pi1val.2 . . 3 (𝜑𝑌𝑋)
21 ovexd 7180 . . 3 (𝜑 → (𝑂 /s ( ≃ph𝐽)) ∈ V)
223, 10, 17, 19, 20, 21ovmpodx 7290 . 2 (𝜑 → (𝐽 π1 𝑌) = (𝑂 /s ( ≃ph𝐽)))
231, 22syl5eq 2865 1 (𝜑𝐺 = (𝑂 /s ( ≃ph𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  Vcvv 3492   cuni 4830  cfv 6348  (class class class)co 7145  cmpo 7147   /s cqus 16766  Topctop 21429  TopOnctopon 21446  phcphtpc 23500   Ω1 comi 23532   π1 cpi1 23534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-topon 21447  df-pi1 23539
This theorem is referenced by:  pi1bas  23569  pi1addf  23578  pi1addval  23579  pi1grplem  23580
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