Theorem ixpf 8927

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 8908	. 2 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))
2		ssiun2 5017	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
3	2	sseld 3939	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) ∈ 𝐵 → (𝐹‘𝑥) ∈ ∪ 𝑥 ∈ 𝐴 𝐵))
4	3	ralimia 3102	. . . . 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 2928	. . . . 5 ⊢ Ⅎ𝑥𝐴
7		nfiu1 4997	. . . . 5 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵
8		nfcv 2928	. . . . 5 ⊢ Ⅎ𝑥𝐹
9	6, 7, 8	ffnfvf 7122	. . . 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 2146  ∀wral 3082  Vcvv 3458  ∪ ciun 4961   Fn wfn 6538  ⟶wf 6539  ‘cfv 6543  Xcixp 8904

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409

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 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ixp 8905

This theorem is used by:  uniixp  8928  ixpssmap2g  8934  ioorrnopnlem  47059  iunhoiioolem  47430