Theorem elixp2 8846

Description: Membership in an infinite Cartesian product. See df-ixp 8843 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 6583	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓 Fn 𝐴 ↔ 𝐹 Fn 𝐴))
2		fveq1 6833	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
3	2	eleq1d 2825	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑥) ∈ 𝐵 ↔ (𝐹‘𝑥) ∈ 𝐵))
4	3	ralbidv 3163	. . . . 5 ⊢ (𝑓 = 𝐹 → (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))
5	1, 4	anbi12d 638	. . . 4 ⊢ (𝑓 = 𝐹 → ((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵) ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)))
6		dfixp 8844	. . . 4 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)}
7	5, 6	elab2g 3625	. . 3 ⊢ (𝐹 ∈ V → (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)))
8	7	pm5.32i 579	. 2 ⊢ ((𝐹 ∈ V ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐵) ↔ (𝐹 ∈ V ∧ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)))
9		elex 3453	. . 3 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝐹 ∈ V)
10	9	pm4.71ri 565	. 2 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐵))
11		3anass 1100	. 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 304	1 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))

Syntax hints:   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3054  Vcvv 3432   Fn wfn 6487  ‘cfv 6492  Xcixp 8842

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6448  df-fun 6494  df-fn 6495  df-fv 6500  df-ixp 8843

This theorem is referenced by:  fvixp  8847  ixpfn  8848  elixp  8849  ixpf  8865  resixp  8878  undifixp  8879  mptelixpg  8880  prdsbasprj  17433  xpsfrnel  17524  xpscf  17527  isssc  17785  isfuncd  17830  funcres2b  17862  dprdw  19985  ptrecube  37994  kelac1  43515  elixpconstg  45543  fvixp2  45652  rrxsnicc  46750  ioorrnopnxrlem  46756  hoiqssbllem1  47072  iinhoiicclem  47123  iunhoiioolem  47125  funcf2lem  49578  isnatd  49720