Theorem ptunimpt 21208

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 2610	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐾) = (𝑥 ∈ 𝐴 ↦ 𝐾)
2	1	fvmpt2 6200	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐾 ∈ Top) → ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑥) = 𝐾)
3	2	eqcomd 2616	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐾 ∈ Top) → 𝐾 = ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑥))
4	3	unieqd 4382	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐾 ∈ Top) → ∪ 𝐾 = ∪ ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑥))
5	4	ralimiaa 2935	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐾 ∈ Top → ∀𝑥 ∈ 𝐴 ∪ 𝐾 = ∪ ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑥))
6	5	adantl 481	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐾 ∈ Top) → ∀𝑥 ∈ 𝐴 ∪ 𝐾 = ∪ ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑥))
7		ixpeq2 7808	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∪ 𝐾 = ∪ ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑥) → X𝑥 ∈ 𝐴 ∪ 𝐾 = X𝑥 ∈ 𝐴 ∪ ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑥))
8	6, 7	syl 17	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐾 ∈ Top) → X𝑥 ∈ 𝐴 ∪ 𝐾 = X𝑥 ∈ 𝐴 ∪ ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑥))
9		nffvmpt1 6111	. . . . 5 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑦)
10	9	nfuni 4378	. . . 4 ⊢ Ⅎ𝑥∪ ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑦)
11		nfcv 2751	. . . 4 ⊢ Ⅎ𝑦∪ ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑥)
12		fveq2 6103	. . . . 5 ⊢ (𝑦 = 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑦) = ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑥))
13	12	unieqd 4382	. . . 4 ⊢ (𝑦 = 𝑥 → ∪ ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑦) = ∪ ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑥))
14	10, 11, 13	cbvixp 7811	. . 3 ⊢ X𝑦 ∈ 𝐴 ∪ ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑦) = X𝑥 ∈ 𝐴 ∪ ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑥)
15	8, 14	syl6eqr 2662	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐾 ∈ Top) → X𝑥 ∈ 𝐴 ∪ 𝐾 = X𝑦 ∈ 𝐴 ∪ ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑦))
16	1	fmpt 6289	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐾 ∈ Top ↔ (𝑥 ∈ 𝐴 ↦ 𝐾):𝐴⟶Top)
17		ptunimpt.j	. . . 4 ⊢ 𝐽 = (∏t‘(𝑥 ∈ 𝐴 ↦ 𝐾))
18	17	ptuni 21207	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐾):𝐴⟶Top) → X𝑦 ∈ 𝐴 ∪ ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑦) = ∪ 𝐽)
19	16, 18	sylan2b 491	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐾 ∈ Top) → X𝑦 ∈ 𝐴 ∪ ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑦) = ∪ 𝐽)
20	15, 19	eqtrd 2644	1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐾 ∈ Top) → X𝑥 ∈ 𝐴 ∪ 𝐾 = ∪ 𝐽)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∪ cuni 4372   ↦ cmpt 4643  ⟶wf 5800  ‘cfv 5804  Xcixp 7794  ∏_tcpt 15922  Topctop 20517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-ixp 7795  df-en 7842  df-fin 7845  df-fi 8200  df-topgen 15927  df-pt 15928  df-top 20521  df-bases 20522

This theorem is referenced by:  pttopon  21209  kelac1  36651