Theorem ixpf 8858

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 8839	. 2 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))
2		ssiun2 5003	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
3	2	sseld 3932	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) ∈ 𝐵 → (𝐹‘𝑥) ∈ ∪ 𝑥 ∈ 𝐴 𝐵))
4	3	ralimia 3070	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
5	4	anim2i 617	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) → (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ ∪ 𝑥 ∈ 𝐴 𝐵))
6		nfcv 2898	. . . . 5 ⊢ Ⅎ𝑥𝐴
7		nfiu1 4982	. . . . 5 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵
8		nfcv 2898	. . . . 5 ⊢ Ⅎ𝑥𝐹
9	6, 7, 8	ffnfvf 7065	. . . 4 ⊢ (𝐹:𝐴⟶∪ 𝑥 ∈ 𝐴 𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ ∪ 𝑥 ∈ 𝐴 𝐵))
10	5, 9	sylibr 234	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) → 𝐹:𝐴⟶∪ 𝑥 ∈ 𝐴 𝐵)
11	10	3adant1 1130	. 2 ⊢ ((𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) → 𝐹:𝐴⟶∪ 𝑥 ∈ 𝐴 𝐵)
12	1, 11	sylbi 217	1 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝐹:𝐴⟶∪ 𝑥 ∈ 𝐴 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2113  ∀wral 3051  Vcvv 3440  ∪ ciun 4946   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  Xcixp 8835

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ixp 8836

This theorem is referenced by:  uniixp  8859  ixpssmap2g  8865  ioorrnopnlem  46544  iunhoiioolem  46915