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| Mirrors > Home > MPE Home > Th. List > fnovrn | Structured version Visualization version GIF version | ||
| Description: An operation's value belongs to its range. (Contributed by NM, 10-Feb-2007.) |
| Ref | Expression |
|---|---|
| fnovrn | ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (𝐶𝐹𝐷) ∈ ran 𝐹) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | opelxpi 5722 | . . 3 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → 〈𝐶, 𝐷〉 ∈ (𝐴 × 𝐵)) | |
| 2 | df-ov 7434 | . . . 4 ⊢ (𝐶𝐹𝐷) = (𝐹‘〈𝐶, 𝐷〉) | |
| 3 | fnfvelrn 7100 | . . . 4 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 〈𝐶, 𝐷〉 ∈ (𝐴 × 𝐵)) → (𝐹‘〈𝐶, 𝐷〉) ∈ ran 𝐹) | |
| 4 | 2, 3 | eqeltrid 2845 | . . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 〈𝐶, 𝐷〉 ∈ (𝐴 × 𝐵)) → (𝐶𝐹𝐷) ∈ ran 𝐹) |
| 5 | 1, 4 | sylan2 593 | . 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → (𝐶𝐹𝐷) ∈ ran 𝐹) |
| 6 | 5 | 3impb 1115 | 1 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (𝐶𝐹𝐷) ∈ ran 𝐹) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 ∈ wcel 2108 〈cop 4632 × cxp 5683 ran crn 5686 Fn wfn 6556 ‘cfv 6561 (class class class)co 7431 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 df-ov 7434 |
| This theorem is referenced by: unirnioo 13489 ioorebas 13491 yonffthlem 18327 gsumval2a 18698 efginvrel2 19745 efgredleme 19761 efgcpbllemb 19773 mplsubrglem 22024 lecldbas 23227 blelrnps 24426 blelrn 24427 blssioo 24816 tgioo 24817 opnmbllem 25636 mbfdm 25661 mbfima 25665 tpr2rico 33911 dya2icoseg 34279 opnmbllem0 37663 elrnmpoid 45233 smflimlem3 46788 |
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