Theorem ixpf 8919

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 8900	. 2 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))
2		ssiun2 5013	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
3	2	sseld 3937	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) ∈ 𝐵 → (𝐹‘𝑥) ∈ ∪ 𝑥 ∈ 𝐴 𝐵))
4	3	ralimia 3099	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
5	4	anim2i 628	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) → (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ ∪ 𝑥 ∈ 𝐴 𝐵))
6		nfcv 2925	. . . . 5 ⊢ Ⅎ𝑥𝐴
7		nfiu1 4993	. . . . 5 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵
8		nfcv 2925	. . . . 5 ⊢ Ⅎ𝑥𝐹
9	6, 7, 8	ffnfvf 7117	. . . 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)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   ∈ wcel 2143  ∀wral 3079  Vcvv 3455  ∪ ciun 4957   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  Xcixp 8896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-ixp 8897

This theorem is referenced by:  uniixp  8920  ixpssmap2g  8926  ioorrnopnlem  47001  iunhoiioolem  47372