Theorem ixpsnval 8873

Description: The value of an infinite Cartesian product with a singleton. (Contributed by AV, 3-Dec-2018.)

Assertion

Ref	Expression
ixpsnval	⊢ (𝑋 ∈ 𝑉 → X𝑥 ∈ {𝑋}𝐵 = {𝑓 ∣ (𝑓 Fn {𝑋} ∧ (𝑓‘𝑋) ∈ ⦋𝑋 / 𝑥⦌𝐵)})

Distinct variable groups:   𝐵,𝑓   𝑓,𝑉   𝑓,𝑋,𝑥

Allowed substitution hints:   𝐵(𝑥)   𝑉(𝑥)

Proof of Theorem ixpsnval

Step	Hyp	Ref	Expression
1		dfixp 8872	. 2 ⊢ X𝑥 ∈ {𝑋}𝐵 = {𝑓 ∣ (𝑓 Fn {𝑋} ∧ ∀𝑥 ∈ {𝑋} (𝑓‘𝑥) ∈ 𝐵)}
2		ralsnsg 4634	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (∀𝑥 ∈ {𝑋} (𝑓‘𝑥) ∈ 𝐵 ↔ [𝑋 / 𝑥](𝑓‘𝑥) ∈ 𝐵))
3		sbcel12 4374	. . . . . 6 ⊢ ([𝑋 / 𝑥](𝑓‘𝑥) ∈ 𝐵 ↔ ⦋𝑋 / 𝑥⦌(𝑓‘𝑥) ∈ ⦋𝑋 / 𝑥⦌𝐵)
4		csbfv2g 6907	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑥⦌(𝑓‘𝑥) = (𝑓‘⦋𝑋 / 𝑥⦌𝑥))
5		csbvarg 4397	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑥⦌𝑥 = 𝑋)
6	5	fveq2d 6862	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → (𝑓‘⦋𝑋 / 𝑥⦌𝑥) = (𝑓‘𝑋))
7	4, 6	eqtrd 2764	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑥⦌(𝑓‘𝑥) = (𝑓‘𝑋))
8	7	eleq1d 2813	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → (⦋𝑋 / 𝑥⦌(𝑓‘𝑥) ∈ ⦋𝑋 / 𝑥⦌𝐵 ↔ (𝑓‘𝑋) ∈ ⦋𝑋 / 𝑥⦌𝐵))
9	3, 8	bitrid 283	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥](𝑓‘𝑥) ∈ 𝐵 ↔ (𝑓‘𝑋) ∈ ⦋𝑋 / 𝑥⦌𝐵))
10	2, 9	bitrd 279	. . . 4 ⊢ (𝑋 ∈ 𝑉 → (∀𝑥 ∈ {𝑋} (𝑓‘𝑥) ∈ 𝐵 ↔ (𝑓‘𝑋) ∈ ⦋𝑋 / 𝑥⦌𝐵))
11	10	anbi2d 630	. . 3 ⊢ (𝑋 ∈ 𝑉 → ((𝑓 Fn {𝑋} ∧ ∀𝑥 ∈ {𝑋} (𝑓‘𝑥) ∈ 𝐵) ↔ (𝑓 Fn {𝑋} ∧ (𝑓‘𝑋) ∈ ⦋𝑋 / 𝑥⦌𝐵)))
12	11	abbidv 2795	. 2 ⊢ (𝑋 ∈ 𝑉 → {𝑓 ∣ (𝑓 Fn {𝑋} ∧ ∀𝑥 ∈ {𝑋} (𝑓‘𝑥) ∈ 𝐵)} = {𝑓 ∣ (𝑓 Fn {𝑋} ∧ (𝑓‘𝑋) ∈ ⦋𝑋 / 𝑥⦌𝐵)})
13	1, 12	eqtrid 2776	1 ⊢ (𝑋 ∈ 𝑉 → X𝑥 ∈ {𝑋}𝐵 = {𝑓 ∣ (𝑓 Fn {𝑋} ∧ (𝑓‘𝑋) ∈ ⦋𝑋 / 𝑥⦌𝐵)})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2707  ∀wral 3044  [wsbc 3753  ⦋csb 3862  {csn 4589   Fn wfn 6506  ‘cfv 6511  Xcixp 8870

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-dm 5648  df-iota 6464  df-fn 6514  df-fv 6519  df-ixp 8871

This theorem is referenced by:  ixpsnbasval  21115