Theorem ixpsnval 8830

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 8829	. 2 ⊢ X𝑥 ∈ {𝑋}𝐵 = {𝑓 ∣ (𝑓 Fn {𝑋} ∧ ∀𝑥 ∈ {𝑋} (𝑓‘𝑥) ∈ 𝐵)}
2		ralsnsg 4622	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (∀𝑥 ∈ {𝑋} (𝑓‘𝑥) ∈ 𝐵 ↔ [𝑋 / 𝑥](𝑓‘𝑥) ∈ 𝐵))
3		sbcel12 4360	. . . . . 6 ⊢ ([𝑋 / 𝑥](𝑓‘𝑥) ∈ 𝐵 ↔ ⦋𝑋 / 𝑥⦌(𝑓‘𝑥) ∈ ⦋𝑋 / 𝑥⦌𝐵)
4		csbfv2g 6874	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑥⦌(𝑓‘𝑥) = (𝑓‘⦋𝑋 / 𝑥⦌𝑥))
5		csbvarg 4383	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑥⦌𝑥 = 𝑋)
6	5	fveq2d 6832	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → (𝑓‘⦋𝑋 / 𝑥⦌𝑥) = (𝑓‘𝑋))
7	4, 6	eqtrd 2768	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑥⦌(𝑓‘𝑥) = (𝑓‘𝑋))
8	7	eleq1d 2818	. . . . . 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 2799	. 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 2780	1 ⊢ (𝑋 ∈ 𝑉 → X𝑥 ∈ {𝑋}𝐵 = {𝑓 ∣ (𝑓 Fn {𝑋} ∧ (𝑓‘𝑋) ∈ ⦋𝑋 / 𝑥⦌𝐵)})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {cab 2711  ∀wral 3048  [wsbc 3737  ⦋csb 3846  {csn 4575   Fn wfn 6481  ‘cfv 6486  Xcixp 8827

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-dm 5629  df-iota 6442  df-fn 6489  df-fv 6494  df-ixp 8828

This theorem is referenced by:  ixpsnbasval  21144