Theorem pi1val 25196

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 25167	. . . 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 782	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → 𝑗 = 𝐽)
5		simprr 784	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → 𝑦 = 𝑌)
6	4, 5	oveq12d 7428	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → (𝑗 Ω1 𝑦) = (𝐽 Ω1 𝑌))
7		pi1val.o	. . . . 5 ⊢ 𝑂 = (𝐽 Ω1 𝑌)
8	6, 7	eqtr4di 2816	. . . 4 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → (𝑗 Ω1 𝑦) = 𝑂)
9	4	fveq2d 6885	. . . 4 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → ( ≃ph‘𝑗) = ( ≃ph‘𝐽))
10	8, 9	oveq12d 7428	. . 3 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → ((𝑗 Ω1 𝑦) /s ( ≃ph‘𝑗)) = (𝑂 /s ( ≃ph‘𝐽)))
11		unieq 4883	. . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
12	11	adantl 486	. . . 4 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → ∪ 𝑗 = ∪ 𝐽)
13		pi1val.1	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
14		toponuni 23071	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
15	13, 14	syl 18	. . . . 5 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
16	15	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → 𝑋 = ∪ 𝐽)
17	12, 16	eqtr4d 2801	. . 3 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → ∪ 𝑗 = 𝑋)
18		topontop 23070	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
19	13, 18	syl 18	. . 3 ⊢ (𝜑 → 𝐽 ∈ Top)
20		pi1val.2	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑋)
21		ovexd 7445	. . 3 ⊢ (𝜑 → (𝑂 /s ( ≃ph‘𝐽)) ∈ V)
22	3, 10, 17, 19, 20, 21	ovmpodx 7561	. 2 ⊢ (𝜑 → (𝐽 π1 𝑌) = (𝑂 /s ( ≃ph‘𝐽)))
23	1, 22	eqtrid 2810	1 ⊢ (𝜑 → 𝐺 = (𝑂 /s ( ≃ph‘𝐽)))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  Vcvv 3455  ∪ cuni 4872  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412   /_s cqus 17554  Topctop 23050  TopOnctopon 23067   ≃_phcphtpc 25128   Ω₁ comi 25160   π₁ cpi1 25162

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-topon 23068  df-pi1 25167

This theorem is referenced by:  pi1bas  25197  pi1addf  25206  pi1addval  25207  pi1grplem  25208