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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fafv2elcdm | Structured version Visualization version GIF version |
Description: An alternate function value belongs to the codomain of the function, analogous to ffvelcdm 7086. (Contributed by AV, 2-Sep-2022.) |
Ref | Expression |
---|---|
fafv2elcdm | ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹''''𝐶) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffn 6717 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
2 | fnafv2elrn 46676 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐹''''𝐶) ∈ ran 𝐹) | |
3 | 1, 2 | sylan 578 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹''''𝐶) ∈ ran 𝐹) |
4 | frn 6724 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵) | |
5 | 4 | sseld 3971 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → ((𝐹''''𝐶) ∈ ran 𝐹 → (𝐹''''𝐶) ∈ 𝐵)) |
6 | 5 | adantr 479 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → ((𝐹''''𝐶) ∈ ran 𝐹 → (𝐹''''𝐶) ∈ 𝐵)) |
7 | 3, 6 | mpd 15 | 1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹''''𝐶) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2098 ran crn 5673 Fn wfn 6538 ⟶wf 6539 ''''cafv2 46651 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-dfat 46562 df-afv2 46652 |
This theorem is referenced by: (None) |
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