Theorem fafv2elrnb 43791

Description: An alternate function value is defined, i.e., belongs to the range of the function, iff its argument is in the domain of the function. (Contributed by AV, 3-Sep-2022.)

Assertion

Ref	Expression
fafv2elrnb	⊢ (𝐹:𝐴⟶𝐵 → (𝐶 ∈ 𝐴 ↔ (𝐹''''𝐶) ∈ ran 𝐹))

Proof of Theorem fafv2elrnb

Step	Hyp	Ref	Expression
1		ffn 6487	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
2		fnafv2elrn 43789	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐹''''𝐶) ∈ ran 𝐹)
3	1, 2	sylan 583	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹''''𝐶) ∈ ran 𝐹)
4	3	ex 416	. 2 ⊢ (𝐹:𝐴⟶𝐵 → (𝐶 ∈ 𝐴 → (𝐹''''𝐶) ∈ ran 𝐹))
5		fdm 6495	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
6		ndmafv2nrn 43778	. . . . . 6 ⊢ (¬ 𝐶 ∈ dom 𝐹 → (𝐹''''𝐶) ∉ ran 𝐹)
7		df-nel 3092	. . . . . 6 ⊢ ((𝐹''''𝐶) ∉ ran 𝐹 ↔ ¬ (𝐹''''𝐶) ∈ ran 𝐹)
8	6, 7	sylib 221	. . . . 5 ⊢ (¬ 𝐶 ∈ dom 𝐹 → ¬ (𝐹''''𝐶) ∈ ran 𝐹)
9	8	con4i 114	. . . 4 ⊢ ((𝐹''''𝐶) ∈ ran 𝐹 → 𝐶 ∈ dom 𝐹)
10		eleq2 2878	. . . 4 ⊢ (dom 𝐹 = 𝐴 → (𝐶 ∈ dom 𝐹 ↔ 𝐶 ∈ 𝐴))
11	9, 10	syl5ib 247	. . 3 ⊢ (dom 𝐹 = 𝐴 → ((𝐹''''𝐶) ∈ ran 𝐹 → 𝐶 ∈ 𝐴))
12	5, 11	syl 17	. 2 ⊢ (𝐹:𝐴⟶𝐵 → ((𝐹''''𝐶) ∈ ran 𝐹 → 𝐶 ∈ 𝐴))
13	4, 12	impbid 215	1 ⊢ (𝐹:𝐴⟶𝐵 → (𝐶 ∈ 𝐴 ↔ (𝐹''''𝐶) ∈ ran 𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2111   ∉ wnel 3091  dom cdm 5519  ran crn 5520   Fn wfn 6319  ⟶wf 6320  ''''cafv2 43764

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-nel 3092  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-dfat 43675  df-afv2 43765

This theorem is referenced by: (None)