Theorem fafv2elrnb 48013

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 6712	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
2		fnafv2elrn 48011	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐹''''𝐶) ∈ ran 𝐹)
3	1, 2	sylan 592	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹''''𝐶) ∈ ran 𝐹)
4	3	ex 418	. 2 ⊢ (𝐹:𝐴⟶𝐵 → (𝐶 ∈ 𝐴 → (𝐹''''𝐶) ∈ ran 𝐹))
5		fdm 6722	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
6		ndmafv2nrn 48000	. . . . . 6 ⊢ (¬ 𝐶 ∈ dom 𝐹 → (𝐹''''𝐶) ∉ ran 𝐹)
7		df-nel 3068	. . . . . 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 115	. . . 4 ⊢ ((𝐹''''𝐶) ∈ ran 𝐹 → 𝐶 ∈ dom 𝐹)
10		eleq2 2855	. . . 4 ⊢ (dom 𝐹 = 𝐴 → (𝐶 ∈ dom 𝐹 ↔ 𝐶 ∈ 𝐴))
11	9, 10	imbitrid 247	. . 3 ⊢ (dom 𝐹 = 𝐴 → ((𝐹''''𝐶) ∈ ran 𝐹 → 𝐶 ∈ 𝐴))
12	5, 11	syl 18	. 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))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2146   ∉ wnel 3067  dom cdm 5666  ran crn 5667   Fn wfn 6538  ⟶wf 6539  ''''cafv2 47986

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-dfat 47897  df-afv2 47987

This theorem is used by: (None)