Theorem afvelrnb 47523

Description: A member of a function's range is a value of the function, analogous to fvelrnb 6902 with the additional requirement that the member must be a set. (Contributed by Alexander van der Vekens, 25-May-2017.)

Assertion

Ref	Expression
afvelrnb	⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝑉) → (𝐵 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem afvelrnb

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnrnafv 47522	. . . 4 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹'''𝑥)})
2	1	adantr 480	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝑉) → ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹'''𝑥)})
3	2	eleq2d 2823	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝑉) → (𝐵 ∈ ran 𝐹 ↔ 𝐵 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹'''𝑥)}))
4		eqeq1 2741	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑦 = (𝐹'''𝑥) ↔ 𝐵 = (𝐹'''𝑥)))
5		eqcom 2744	. . . . . 6 ⊢ (𝐵 = (𝐹'''𝑥) ↔ (𝐹'''𝑥) = 𝐵)
6	4, 5	bitrdi 287	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 = (𝐹'''𝑥) ↔ (𝐹'''𝑥) = 𝐵))
7	6	rexbidv 3162	. . . 4 ⊢ (𝑦 = 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 = (𝐹'''𝑥) ↔ ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝐵))
8	7	elabg 3633	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹'''𝑥)} ↔ ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝐵))
9	8	adantl 481	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝑉) → (𝐵 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹'''𝑥)} ↔ ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝐵))
10	3, 9	bitrd 279	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝑉) → (𝐵 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2715  ∃wrex 3062  ran crn 5633   Fn wfn 6495  '''cafv 47477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-iota 6456  df-fun 6502  df-fn 6503  df-fv 6508  df-aiota 47445  df-dfat 47479  df-afv 47480

This theorem is referenced by: (None)