Theorem ixpsnval 6698

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 6697	. 2 ⊢ X𝑥 ∈ {𝑋}𝐵 = {𝑓 ∣ (𝑓 Fn {𝑋} ∧ ∀𝑥 ∈ {𝑋} (𝑓‘𝑥) ∈ 𝐵)}
2		ralsnsg 3629	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (∀𝑥 ∈ {𝑋} (𝑓‘𝑥) ∈ 𝐵 ↔ [𝑋 / 𝑥](𝑓‘𝑥) ∈ 𝐵))
3		sbcel12g 3072	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥](𝑓‘𝑥) ∈ 𝐵 ↔ ⦋𝑋 / 𝑥⦌(𝑓‘𝑥) ∈ ⦋𝑋 / 𝑥⦌𝐵))
4		csbfvg 5552	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑥⦌(𝑓‘𝑥) = (𝑓‘𝑋))
5	4	eleq1d 2246	. . . . 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 2295	. 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 2222	1 ⊢ (𝑋 ∈ 𝑉 → X𝑥 ∈ {𝑋}𝐵 = {𝑓 ∣ (𝑓 Fn {𝑋} ∧ (𝑓‘𝑋) ∈ ⦋𝑋 / 𝑥⦌𝐵)})

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1353   ∈ wcel 2148  {cab 2163  ∀wral 2455  [wsbc 2962  ⦋csb 3057  {csn 3592   Fn wfn 5210  ‘cfv 5215  Xcixp 6695

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-iota 5177  df-fn 5218  df-fv 5223  df-ixp 6696

This theorem is referenced by: (None)