Theorem ixpf 8979

Description: A member of an infinite Cartesian product maps to the indexed union of the product argument. Remark in [Enderton] p. 54. (Contributed by NM, 28-Sep-2006.)

Assertion

Ref	Expression
ixpf	⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝐹:𝐴⟶∪ 𝑥 ∈ 𝐴 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem ixpf

Step	Hyp	Ref	Expression
1		elixp2 8960	. 2 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))
2		ssiun2 5009	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
3	2	sseld 3912	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) ∈ 𝐵 → (𝐹‘𝑥) ∈ ∪ 𝑥 ∈ 𝐴 𝐵))
4	3	ralimia 3100	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
5	4	anim2i 629	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) → (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ ∪ 𝑥 ∈ 𝐴 𝐵))
6		nfcv 2926	. . . . 5 ⊢ Ⅎ𝑥𝐴
7		nfiu1 4989	. . . . 5 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵
8		nfcv 2926	. . . . 5 ⊢ Ⅎ𝑥𝐹
9	6, 7, 8	ffnfvf 7147	. . . 4 ⊢ (𝐹:𝐴⟶∪ 𝑥 ∈ 𝐴 𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ ∪ 𝑥 ∈ 𝐴 𝐵))
10	5, 9	sylibr 237	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) → 𝐹:𝐴⟶∪ 𝑥 ∈ 𝐴 𝐵)
11	10	3adant1 1148	. 2 ⊢ ((𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) → 𝐹:𝐴⟶∪ 𝑥 ∈ 𝐴 𝐵)
12	1, 11	sylbi 220	1 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝐹:𝐴⟶∪ 𝑥 ∈ 𝐴 𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   ∈ wcel 2147  ∀wral 3080  Vcvv 3454  ∪ ciun 4954   Fn wfn 6556  ⟶wf 6557  ‘cfv 6561  Xcixp 8956

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ixp 8957

This theorem is used by:  uniixp  8980  ixpssmap2g  8986  ioorrnopnlem  47529  iunhoiioolem  47900