Theorem elixp2 6950

Description: Membership in an infinite Cartesian product. See df-ixp 6947 for discussion of the notation. (Contributed by NM, 28-Sep-2006.)

Assertion

Ref	Expression
elixp2	⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem elixp2

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fneq1 5449	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓 Fn 𝐴 ↔ 𝐹 Fn 𝐴))
2		fveq1 5674	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
3	2	eleq1d 2303	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑥) ∈ 𝐵 ↔ (𝐹‘𝑥) ∈ 𝐵))
4	3	ralbidv 2544	. . . . 5 ⊢ (𝑓 = 𝐹 → (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))
5	1, 4	anbi12d 473	. . . 4 ⊢ (𝑓 = 𝐹 → ((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵) ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)))
6		dfixp 6948	. . . 4 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)}
7	5, 6	elab2g 2967	. . 3 ⊢ (𝐹 ∈ V → (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)))
8	7	pm5.32i 454	. 2 ⊢ ((𝐹 ∈ V ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐵) ↔ (𝐹 ∈ V ∧ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)))
9		elex 2827	. . 3 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝐹 ∈ V)
10	9	pm4.71ri 392	. 2 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐵))
11		3anass 1009	. 2 ⊢ ((𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) ↔ (𝐹 ∈ V ∧ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)))
12	8, 10, 11	3bitr4i 212	1 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))

Syntax hints:   ∧ wa 104   ↔ wb 105   ∧ w3a 1005   = wceq 1398   ∈ wcel 2205  ∀wral 2522  Vcvv 2815   Fn wfn 5352  ‘cfv 5357  Xcixp 6946

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365  df-ixp 6947

This theorem is referenced by:  fvixp  6951  ixpfn  6952  elixp  6953  ixpf  6968  resixp  6981  mptelixpg  6982  prdsbasprj  13579  xpsfrnel  13641  xpscf  13644  depindlem2  16614