Theorem pi1val 25084

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 25055	. . . 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 771	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → 𝑗 = 𝐽)
5		simprr 773	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → 𝑦 = 𝑌)
6	4, 5	oveq12d 7449	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → (𝑗 Ω1 𝑦) = (𝐽 Ω1 𝑌))
7		pi1val.o	. . . . 5 ⊢ 𝑂 = (𝐽 Ω1 𝑌)
8	6, 7	eqtr4di 2793	. . . 4 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → (𝑗 Ω1 𝑦) = 𝑂)
9	4	fveq2d 6911	. . . 4 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → ( ≃ph‘𝑗) = ( ≃ph‘𝐽))
10	8, 9	oveq12d 7449	. . 3 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → ((𝑗 Ω1 𝑦) /s ( ≃ph‘𝑗)) = (𝑂 /s ( ≃ph‘𝐽)))
11		unieq 4923	. . . . 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 22936	. . . . . 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 2778	. . 3 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → ∪ 𝑗 = 𝑋)
18		topontop 22935	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
19	13, 18	syl 17	. . 3 ⊢ (𝜑 → 𝐽 ∈ Top)
20		pi1val.2	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑋)
21		ovexd 7466	. . 3 ⊢ (𝜑 → (𝑂 /s ( ≃ph‘𝐽)) ∈ V)
22	3, 10, 17, 19, 20, 21	ovmpodx 7584	. 2 ⊢ (𝜑 → (𝐽 π1 𝑌) = (𝑂 /s ( ≃ph‘𝐽)))
23	1, 22	eqtrid 2787	1 ⊢ (𝜑 → 𝐺 = (𝑂 /s ( ≃ph‘𝐽)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  Vcvv 3478  ∪ cuni 4912  ‘cfv 6563  (class class class)co 7431   ∈ cmpo 7433   /_s cqus 17552  Topctop 22915  TopOnctopon 22932   ≃_phcphtpc 25015   Ω₁ comi 25048   π₁ cpi1 25050

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-topon 22933  df-pi1 25055

This theorem is referenced by:  pi1bas  25085  pi1addf  25094  pi1addval  25095  pi1grplem  25096