Theorem afvelrnb0 46535

Description: A member of a function's range is a value of the function, only one direction of implication of fvelrnb 6954. (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 46533	. . 3 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹'''𝑥)})
2	1	eleq2d 2815	. 2 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 ↔ 𝐵 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹'''𝑥)}))
3		eqeq1 2732	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑦 = (𝐹'''𝑥) ↔ 𝐵 = (𝐹'''𝑥)))
4		eqcom 2735	. . . . . 6 ⊢ (𝐵 = (𝐹'''𝑥) ↔ (𝐹'''𝑥) = 𝐵)
5	3, 4	bitrdi 287	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 = (𝐹'''𝑥) ↔ (𝐹'''𝑥) = 𝐵))
6	5	rexbidv 3174	. . . 4 ⊢ (𝑦 = 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 = (𝐹'''𝑥) ↔ ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝐵))
7	6	elabg 3664	. . 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	syl6bi 253	1 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 → ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝐵))

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099  {cab 2705  ∃wrex 3066  ran crn 5674   Fn wfn 6538  '''cafv 46488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pr 5424

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-int 4946  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551  df-aiota 46456  df-dfat 46490  df-afv 46491

This theorem is referenced by:  ffnafv  46542