Theorem pi1val 24954

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	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
pi1val.2	⊢ (𝜑 → 𝑌 ∈ 𝑋)
pi1val.o	⊢ 𝑂 = (𝐽 Ω1 𝑌)

Assertion

Ref	Expression
pi1val	⊢ (𝜑 → 𝐺 = (𝑂 /s ( ≃ph‘𝐽)))

Proof of Theorem pi1val

Dummy variables 𝑗 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pi1val.g	. 2 ⊢ 𝐺 = (𝐽 π1 𝑌)
2		df-pi1 24925	. . . 4 ⊢ π1 = (𝑗 ∈ Top, 𝑦 ∈ ∪ 𝑗 ↦ ((𝑗 Ω1 𝑦) /s ( ≃ph‘𝑗)))
3	2	a1i 11	. . 3 ⊢ (𝜑 → π1 = (𝑗 ∈ Top, 𝑦 ∈ ∪ 𝑗 ↦ ((𝑗 Ω1 𝑦) /s ( ≃ph‘𝑗))))
4		simprl 770	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → 𝑗 = 𝐽)
5		simprr 772	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → 𝑦 = 𝑌)
6	4, 5	oveq12d 7371	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → (𝑗 Ω1 𝑦) = (𝐽 Ω1 𝑌))
7		pi1val.o	. . . . 5 ⊢ 𝑂 = (𝐽 Ω1 𝑌)
8	6, 7	eqtr4di 2782	. . . 4 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → (𝑗 Ω1 𝑦) = 𝑂)
9	4	fveq2d 6830	. . . 4 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → ( ≃ph‘𝑗) = ( ≃ph‘𝐽))
10	8, 9	oveq12d 7371	. . 3 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → ((𝑗 Ω1 𝑦) /s ( ≃ph‘𝑗)) = (𝑂 /s ( ≃ph‘𝐽)))
11		unieq 4872	. . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
12	11	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → ∪ 𝑗 = ∪ 𝐽)
13		pi1val.1	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
14		toponuni 22818	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
15	13, 14	syl 17	. . . . 5 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
16	15	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → 𝑋 = ∪ 𝐽)
17	12, 16	eqtr4d 2767	. . 3 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → ∪ 𝑗 = 𝑋)
18		topontop 22817	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
19	13, 18	syl 17	. . 3 ⊢ (𝜑 → 𝐽 ∈ Top)
20		pi1val.2	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑋)
21		ovexd 7388	. . 3 ⊢ (𝜑 → (𝑂 /s ( ≃ph‘𝐽)) ∈ V)
22	3, 10, 17, 19, 20, 21	ovmpodx 7504	. 2 ⊢ (𝜑 → (𝐽 π1 𝑌) = (𝑂 /s ( ≃ph‘𝐽)))
23	1, 22	eqtrid 2776	1 ⊢ (𝜑 → 𝐺 = (𝑂 /s ( ≃ph‘𝐽)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3438  ∪ cuni 4861  ‘cfv 6486  (class class class)co 7353   ∈ cmpo 7355   /_s cqus 17428  Topctop 22797  TopOnctopon 22814   ≃_phcphtpc 24885   Ω₁ comi 24918   π₁ cpi1 24920

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-topon 22815  df-pi1 24925

This theorem is referenced by:  pi1bas  24955  pi1addf  24964  pi1addval  24965  pi1grplem  24966