Theorem afvelrnb0 47165

Description: A member of a function's range is a value of the function, only one direction of implication of fvelrnb 6921. (Contributed by Alexander van der Vekens, 1-Jun-2017.)

Assertion

Ref	Expression
afvelrnb0	⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 → ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝐵))

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

Proof of Theorem afvelrnb0

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnrnafv 47163	. . 3 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹'''𝑥)})
2	1	eleq2d 2814	. 2 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 ↔ 𝐵 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹'''𝑥)}))
3		eqeq1 2733	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑦 = (𝐹'''𝑥) ↔ 𝐵 = (𝐹'''𝑥)))
4		eqcom 2736	. . . . . 6 ⊢ (𝐵 = (𝐹'''𝑥) ↔ (𝐹'''𝑥) = 𝐵)
5	3, 4	bitrdi 287	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 = (𝐹'''𝑥) ↔ (𝐹'''𝑥) = 𝐵))
6	5	rexbidv 3157	. . . 4 ⊢ (𝑦 = 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 = (𝐹'''𝑥) ↔ ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝐵))
7	6	elabg 3643	. . 3 ⊢ (𝐵 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹'''𝑥)} → (𝐵 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹'''𝑥)} ↔ ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝐵))
8	7	ibi 267	. 2 ⊢ (𝐵 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹'''𝑥)} → ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝐵)
9	2, 8	biimtrdi 253	1 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 → ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝐵))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  {cab 2707  ∃wrex 3053  ran crn 5639   Fn wfn 6506  '''cafv 47118

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-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  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-res 5650  df-iota 6464  df-fun 6513  df-fn 6514  df-fv 6519  df-aiota 47086  df-dfat 47120  df-afv 47121

This theorem is referenced by:  ffnafv  47172