Theorem ixpsnval 6798

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 6797	. 2 ⊢ X𝑥 ∈ {𝑋}𝐵 = {𝑓 ∣ (𝑓 Fn {𝑋} ∧ ∀𝑥 ∈ {𝑋} (𝑓‘𝑥) ∈ 𝐵)}
2		ralsnsg 3672	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (∀𝑥 ∈ {𝑋} (𝑓‘𝑥) ∈ 𝐵 ↔ [𝑋 / 𝑥](𝑓‘𝑥) ∈ 𝐵))
3		sbcel12g 3110	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥](𝑓‘𝑥) ∈ 𝐵 ↔ ⦋𝑋 / 𝑥⦌(𝑓‘𝑥) ∈ ⦋𝑋 / 𝑥⦌𝐵))
4		csbfvg 5626	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑥⦌(𝑓‘𝑥) = (𝑓‘𝑋))
5	4	eleq1d 2275	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (⦋𝑋 / 𝑥⦌(𝑓‘𝑥) ∈ ⦋𝑋 / 𝑥⦌𝐵 ↔ (𝑓‘𝑋) ∈ ⦋𝑋 / 𝑥⦌𝐵))
6	2, 3, 5	3bitrd 214	. . . 4 ⊢ (𝑋 ∈ 𝑉 → (∀𝑥 ∈ {𝑋} (𝑓‘𝑥) ∈ 𝐵 ↔ (𝑓‘𝑋) ∈ ⦋𝑋 / 𝑥⦌𝐵))
7	6	anbi2d 464	. . 3 ⊢ (𝑋 ∈ 𝑉 → ((𝑓 Fn {𝑋} ∧ ∀𝑥 ∈ {𝑋} (𝑓‘𝑥) ∈ 𝐵) ↔ (𝑓 Fn {𝑋} ∧ (𝑓‘𝑋) ∈ ⦋𝑋 / 𝑥⦌𝐵)))
8	7	abbidv 2324	. 2 ⊢ (𝑋 ∈ 𝑉 → {𝑓 ∣ (𝑓 Fn {𝑋} ∧ ∀𝑥 ∈ {𝑋} (𝑓‘𝑥) ∈ 𝐵)} = {𝑓 ∣ (𝑓 Fn {𝑋} ∧ (𝑓‘𝑋) ∈ ⦋𝑋 / 𝑥⦌𝐵)})
9	1, 8	eqtrid 2251	1 ⊢ (𝑋 ∈ 𝑉 → X𝑥 ∈ {𝑋}𝐵 = {𝑓 ∣ (𝑓 Fn {𝑋} ∧ (𝑓‘𝑋) ∈ ⦋𝑋 / 𝑥⦌𝐵)})

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373   ∈ wcel 2177  {cab 2192  ∀wral 2485  [wsbc 3000  ⦋csb 3095  {csn 3635   Fn wfn 5272  ‘cfv 5277  Xcixp 6795

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-sbc 3001  df-csb 3096  df-un 3172  df-sn 3641  df-pr 3642  df-op 3644  df-uni 3854  df-br 4049  df-iota 5238  df-fn 5280  df-fv 5285  df-ixp 6796

This theorem is referenced by:  ixpsnbasval  14278