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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 5719 | . . 3 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → 〈𝐶, 𝐷〉 ∈ (𝐴 × 𝐵)) | |
2 | df-ov 7429 | . . . 4 ⊢ (𝐶𝐹𝐷) = (𝐹‘〈𝐶, 𝐷〉) | |
3 | fnfvelrn 7095 | . . . 4 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 〈𝐶, 𝐷〉 ∈ (𝐴 × 𝐵)) → (𝐹‘〈𝐶, 𝐷〉) ∈ ran 𝐹) | |
4 | 2, 3 | eqeltrid 2833 | . . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 〈𝐶, 𝐷〉 ∈ (𝐴 × 𝐵)) → (𝐶𝐹𝐷) ∈ ran 𝐹) |
5 | 1, 4 | sylan2 591 | . 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → (𝐶𝐹𝐷) ∈ ran 𝐹) |
6 | 5 | 3impb 1112 | 1 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (𝐶𝐹𝐷) ∈ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 ∈ wcel 2098 〈cop 4638 × cxp 5680 ran crn 5683 Fn wfn 6548 ‘cfv 6553 (class class class)co 7426 |
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 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-iota 6505 df-fun 6555 df-fn 6556 df-fv 6561 df-ov 7429 |
This theorem is referenced by: unirnioo 13466 ioorebas 13468 yonffthlem 18281 gsumval2a 18652 efginvrel2 19689 efgredleme 19705 efgcpbllemb 19717 mplsubrglem 21953 lecldbas 23143 blelrnps 24342 blelrn 24343 blssioo 24731 tgioo 24732 opnmbllem 25550 mbfdm 25575 mbfima 25579 tpr2rico 33546 dya2icoseg 33930 opnmbllem0 37162 elrnmpoid 44631 smflimlem3 46190 |
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