Theorem afvelrnb0 47718

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 47716	. . 3 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹'''𝑥)})
2	1	eleq2d 2847	. 2 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 ↔ 𝐵 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹'''𝑥)}))
3		eqeq1 2765	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑦 = (𝐹'''𝑥) ↔ 𝐵 = (𝐹'''𝑥)))
4		eqcom 2768	. . . . . 6 ⊢ (𝐵 = (𝐹'''𝑥) ↔ (𝐹'''𝑥) = 𝐵)
5	3, 4	bitrdi 289	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 = (𝐹'''𝑥) ↔ (𝐹'''𝑥) = 𝐵))
6	5	rexbidv 3185	. . . 4 ⊢ (𝑦 = 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 = (𝐹'''𝑥) ↔ ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝐵))
7	6	elabg 3634	. . 3 ⊢ (𝐵 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹'''𝑥)} → (𝐵 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹'''𝑥)} ↔ ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝐵))
8	7	ibi 269	. 2 ⊢ (𝐵 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹'''𝑥)} → ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝐵)
9	2, 8	biimtrdi 255	1 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 → ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝐵))

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141  {cab 2739  ∃wrex 3085  ran crn 5644   Fn wfn 6510  '''cafv 47671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4903  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-iota 6471  df-fun 6517  df-fn 6518  df-fv 6523  df-aiota 47639  df-dfat 47673  df-afv 47674

This theorem is referenced by:  ffnafv  47725