Theorem ptunimpt 23723

Description: Base set of a product topology given by substitution. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Hypothesis

Ref	Expression
ptunimpt.j	⊢ 𝐽 = (∏t‘(𝑥 ∈ 𝐴 ↦ 𝐾))

Assertion

Ref	Expression
ptunimpt	⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐾 ∈ Top) → X𝑥 ∈ 𝐴 ∪ 𝐾 = ∪ 𝐽)

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝐽(𝑥)   𝐾(𝑥)   𝑉(𝑥)

Proof of Theorem ptunimpt

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2769	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐾) = (𝑥 ∈ 𝐴 ↦ 𝐾)
2	1	fvmpt2 7004	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐾 ∈ Top) → ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑥) = 𝐾)
3	2	eqcomd 2775	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐾 ∈ Top) → 𝐾 = ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑥))
4	3	unieqd 4889	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐾 ∈ Top) → ∪ 𝐾 = ∪ ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑥))
5	4	ralimiaa 3107	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐾 ∈ Top → ∀𝑥 ∈ 𝐴 ∪ 𝐾 = ∪ ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑥))
6	5	adantl 486	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐾 ∈ Top) → ∀𝑥 ∈ 𝐴 ∪ 𝐾 = ∪ ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑥))
7		ixpeq2 8911	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∪ 𝐾 = ∪ ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑥) → X𝑥 ∈ 𝐴 ∪ 𝐾 = X𝑥 ∈ 𝐴 ∪ ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑥))
8	6, 7	syl 18	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐾 ∈ Top) → X𝑥 ∈ 𝐴 ∪ 𝐾 = X𝑥 ∈ 𝐴 ∪ ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑥))
9		nffvmpt1 6895	. . . . 5 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑦)
10	9	nfuni 4883	. . . 4 ⊢ Ⅎ𝑥∪ ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑦)
11		nfcv 2931	. . . 4 ⊢ Ⅎ𝑦∪ ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑥)
12		fveq2 6884	. . . . 5 ⊢ (𝑦 = 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑦) = ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑥))
13	12	unieqd 4889	. . . 4 ⊢ (𝑦 = 𝑥 → ∪ ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑦) = ∪ ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑥))
14	10, 11, 13	cbvixp 8914	. . 3 ⊢ X𝑦 ∈ 𝐴 ∪ ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑦) = X𝑥 ∈ 𝐴 ∪ ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑥)
15	8, 14	eqtr4di 2822	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐾 ∈ Top) → X𝑥 ∈ 𝐴 ∪ 𝐾 = X𝑦 ∈ 𝐴 ∪ ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑦))
16	1	fmpt 7108	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐾 ∈ Top ↔ (𝑥 ∈ 𝐴 ↦ 𝐾):𝐴⟶Top)
17		ptunimpt.j	. . . 4 ⊢ 𝐽 = (∏t‘(𝑥 ∈ 𝐴 ↦ 𝐾))
18	17	ptuni 23722	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐾):𝐴⟶Top) → X𝑦 ∈ 𝐴 ∪ ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑦) = ∪ 𝐽)
19	16, 18	sylan2b 605	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐾 ∈ Top) → X𝑦 ∈ 𝐴 ∪ ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑦) = ∪ 𝐽)
20	15, 19	eqtrd 2804	1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐾 ∈ Top) → X𝑥 ∈ 𝐴 ∪ 𝐾 = ∪ 𝐽)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∪ cuni 4876   ↦ cmpt 5196  ⟶wf 6535  ‘cfv 6539  Xcixp 8897  ∏_tcpt 17493  Topctop 23021

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5273  ax-pow 5339  ax-pr 5407  ax-un 7735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5559  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-xp 5670  df-rel 5671  df-cnv 5672  df-co 5673  df-dm 5674  df-rn 5675  df-res 5676  df-ima 5677  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6495  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-om 7865  df-1o 8455  df-2o 8456  df-ixp 8898  df-en 8946  df-fin 8949  df-fi 9373  df-topgen 17498  df-pt 17499  df-top 23022  df-bases 23074

This theorem is referenced by:  pttopon  23724  kelac1  43719