Theorem kqffn 21338

Description: The topological indistinguishability map is a function on the base. (Contributed by Mario Carneiro, 25-Aug-2015.)

Hypothesis

Ref	Expression
kqval.2	⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})

Assertion

Ref	Expression
kqffn	⊢ (𝐽 ∈ 𝑉 → 𝐹 Fn 𝑋)

Distinct variable groups:   𝑥,𝑦,𝐽   𝑥,𝑋,𝑦   𝑥,𝑉

Allowed substitution hints:   𝐹(𝑥,𝑦)   𝑉(𝑦)

Proof of Theorem kqffn

Step	Hyp	Ref	Expression
1		ssrab2 3650	. . . . 5 ⊢ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦} ⊆ 𝐽
2		elpw2g 4754	. . . . 5 ⊢ (𝐽 ∈ 𝑉 → ({𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦} ∈ 𝒫 𝐽 ↔ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦} ⊆ 𝐽))
3	1, 2	mpbiri 247	. . . 4 ⊢ (𝐽 ∈ 𝑉 → {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦} ∈ 𝒫 𝐽)
4	3	adantr 480	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝑥 ∈ 𝑋) → {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦} ∈ 𝒫 𝐽)
5		kqval.2	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
6	4, 5	fmptd 6292	. 2 ⊢ (𝐽 ∈ 𝑉 → 𝐹:𝑋⟶𝒫 𝐽)
7		ffn 5958	. 2 ⊢ (𝐹:𝑋⟶𝒫 𝐽 → 𝐹 Fn 𝑋)
8	6, 7	syl 17	1 ⊢ (𝐽 ∈ 𝑉 → 𝐹 Fn 𝑋)

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  {crab 2900   ⊆ wss 3540  𝒫 cpw 4108   ↦ cmpt 4643   Fn wfn 5799  ⟶wf 5800

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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  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-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  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-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812

This theorem is referenced by:  kqtopon  21340  kqid  21341  ist0-4  21342  kqfvima  21343  kqsat  21344  kqdisj  21345  kqcldsat  21346  kqopn  21347  kqcld  21348  kqt0lem  21349  isr0  21350  r0cld  21351  regr1lem2  21353  kqreglem1  21354  kqreglem2  21355  kqnrmlem1  21356  kqnrmlem2  21357