Theorem pi1val 25089

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 25060	. . . 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 7466	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → (𝑗 Ω1 𝑦) = (𝐽 Ω1 𝑌))
7		pi1val.o	. . . . 5 ⊢ 𝑂 = (𝐽 Ω1 𝑌)
8	6, 7	eqtr4di 2798	. . . 4 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → (𝑗 Ω1 𝑦) = 𝑂)
9	4	fveq2d 6924	. . . 4 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → ( ≃ph‘𝑗) = ( ≃ph‘𝐽))
10	8, 9	oveq12d 7466	. . 3 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → ((𝑗 Ω1 𝑦) /s ( ≃ph‘𝑗)) = (𝑂 /s ( ≃ph‘𝐽)))
11		unieq 4942	. . . . 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 22941	. . . . . 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 2783	. . 3 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → ∪ 𝑗 = 𝑋)
18		topontop 22940	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
19	13, 18	syl 17	. . 3 ⊢ (𝜑 → 𝐽 ∈ Top)
20		pi1val.2	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑋)
21		ovexd 7483	. . 3 ⊢ (𝜑 → (𝑂 /s ( ≃ph‘𝐽)) ∈ V)
22	3, 10, 17, 19, 20, 21	ovmpodx 7601	. 2 ⊢ (𝜑 → (𝐽 π1 𝑌) = (𝑂 /s ( ≃ph‘𝐽)))
23	1, 22	eqtrid 2792	1 ⊢ (𝜑 → 𝐺 = (𝑂 /s ( ≃ph‘𝐽)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  Vcvv 3488  ∪ cuni 4931  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450   /_s cqus 17565  Topctop 22920  TopOnctopon 22937   ≃_phcphtpc 25020   Ω₁ comi 25053   π₁ cpi1 25055

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-topon 22938  df-pi1 25060

This theorem is referenced by:  pi1bas  25090  pi1addf  25099  pi1addval  25100  pi1grplem  25101