Theorem ixpsnval 8939

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 8938	. 2 ⊢ X𝑥 ∈ {𝑋}𝐵 = {𝑓 ∣ (𝑓 Fn {𝑋} ∧ ∀𝑥 ∈ {𝑋} (𝑓‘𝑥) ∈ 𝐵)}
2		ralsnsg 4675	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (∀𝑥 ∈ {𝑋} (𝑓‘𝑥) ∈ 𝐵 ↔ [𝑋 / 𝑥](𝑓‘𝑥) ∈ 𝐵))
3		sbcel12 4417	. . . . . 6 ⊢ ([𝑋 / 𝑥](𝑓‘𝑥) ∈ 𝐵 ↔ ⦋𝑋 / 𝑥⦌(𝑓‘𝑥) ∈ ⦋𝑋 / 𝑥⦌𝐵)
4		csbfv2g 6956	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑥⦌(𝑓‘𝑥) = (𝑓‘⦋𝑋 / 𝑥⦌𝑥))
5		csbvarg 4440	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑥⦌𝑥 = 𝑋)
6	5	fveq2d 6911	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → (𝑓‘⦋𝑋 / 𝑥⦌𝑥) = (𝑓‘𝑋))
7	4, 6	eqtrd 2775	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑥⦌(𝑓‘𝑥) = (𝑓‘𝑋))
8	7	eleq1d 2824	. . . . . 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 2806	. 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 2787	1 ⊢ (𝑋 ∈ 𝑉 → X𝑥 ∈ {𝑋}𝐵 = {𝑓 ∣ (𝑓 Fn {𝑋} ∧ (𝑓‘𝑋) ∈ ⦋𝑋 / 𝑥⦌𝐵)})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  {cab 2712  ∀wral 3059  [wsbc 3791  ⦋csb 3908  {csn 4631   Fn wfn 6558  ‘cfv 6563  Xcixp 8936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-dm 5699  df-iota 6516  df-fn 6566  df-fv 6571  df-ixp 8937

This theorem is referenced by:  ixpsnbasval  21233