Theorem chfnrn 7021

Description: The range of a choice function (a function that chooses an element from each member of its domain) is included in the union of its domain. (Contributed by NM, 31-Aug-1999.)

Assertion

Ref	Expression
chfnrn	⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝑥) → ran 𝐹 ⊆ ∪ 𝐴)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem chfnrn

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvelrnb 6921	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦))
2	1	biimpd 229	. . . 4 ⊢ (𝐹 Fn 𝐴 → (𝑦 ∈ ran 𝐹 → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦))
3		eleq1 2816	. . . . . . 7 ⊢ ((𝐹‘𝑥) = 𝑦 → ((𝐹‘𝑥) ∈ 𝑥 ↔ 𝑦 ∈ 𝑥))
4	3	biimpcd 249	. . . . . 6 ⊢ ((𝐹‘𝑥) ∈ 𝑥 → ((𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝑥))
5	4	ralimi 3066	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝑥 → ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝑥))
6		rexim 3070	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝑥) → (∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦 → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥))
7	5, 6	syl 17	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝑥 → (∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦 → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥))
8	2, 7	sylan9 507	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝑥) → (𝑦 ∈ ran 𝐹 → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥))
9		eluni2 4875	. . 3 ⊢ (𝑦 ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
10	8, 9	imbitrrdi 252	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝑥) → (𝑦 ∈ ran 𝐹 → 𝑦 ∈ ∪ 𝐴))
11	10	ssrdv 3952	1 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝑥) → ran 𝐹 ⊆ ∪ 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   ⊆ wss 3914  ∪ cuni 4871  ran crn 5639   Fn wfn 6506  ‘cfv 6511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6464  df-fun 6513  df-fn 6514  df-fv 6519

This theorem is referenced by:  stoweidlem59  46057