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Theorem pi1val 22645
 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 22616 . . . 4 π1 = (𝑗 ∈ Top, 𝑦 𝑗 ↦ ((𝑗 Ω1 𝑦) /s ( ≃ph𝑗)))
32a1i 11 . . 3 (𝜑π1 = (𝑗 ∈ Top, 𝑦 𝑗 ↦ ((𝑗 Ω1 𝑦) /s ( ≃ph𝑗))))
4 simprl 790 . . . . . 6 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → 𝑗 = 𝐽)
5 simprr 792 . . . . . 6 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → 𝑦 = 𝑌)
64, 5oveq12d 6567 . . . . 5 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → (𝑗 Ω1 𝑦) = (𝐽 Ω1 𝑌))
7 pi1val.o . . . . 5 𝑂 = (𝐽 Ω1 𝑌)
86, 7syl6eqr 2662 . . . 4 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → (𝑗 Ω1 𝑦) = 𝑂)
94fveq2d 6107 . . . 4 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → ( ≃ph𝑗) = ( ≃ph𝐽))
108, 9oveq12d 6567 . . 3 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → ((𝑗 Ω1 𝑦) /s ( ≃ph𝑗)) = (𝑂 /s ( ≃ph𝐽)))
11 unieq 4380 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝐽)
1211adantl 481 . . . 4 ((𝜑𝑗 = 𝐽) → 𝑗 = 𝐽)
13 pi1val.1 . . . . . 6 (𝜑𝐽 ∈ (TopOn‘𝑋))
14 toponuni 20542 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
1513, 14syl 17 . . . . 5 (𝜑𝑋 = 𝐽)
1615adantr 480 . . . 4 ((𝜑𝑗 = 𝐽) → 𝑋 = 𝐽)
1712, 16eqtr4d 2647 . . 3 ((𝜑𝑗 = 𝐽) → 𝑗 = 𝑋)
18 topontop 20541 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
1913, 18syl 17 . . 3 (𝜑𝐽 ∈ Top)
20 pi1val.2 . . 3 (𝜑𝑌𝑋)
21 ovex 6577 . . . 4 (𝑂 /s ( ≃ph𝐽)) ∈ V
2221a1i 11 . . 3 (𝜑 → (𝑂 /s ( ≃ph𝐽)) ∈ V)
233, 10, 17, 19, 20, 22ovmpt2dx 6685 . 2 (𝜑 → (𝐽 π1 𝑌) = (𝑂 /s ( ≃ph𝐽)))
241, 23syl5eq 2656 1 (𝜑𝐺 = (𝑂 /s ( ≃ph𝐽)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ∪ cuni 4372  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551   /s cqus 15988  Topctop 20517  TopOnctopon 20518   ≃phcphtpc 22576   Ω1 comi 22609   π1 cpi1 22611 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-topon 20523  df-pi1 22616 This theorem is referenced by:  pi1bas  22646  pi1addf  22655  pi1addval  22656  pi1grplem  22657
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