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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 5726 | . . 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 2843 | . . 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 1114 | 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 1086 ∈ wcel 2106 〈cop 4637 × cxp 5687 ran crn 5690 Fn wfn 6558 ‘cfv 6563 (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 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fn 6566 df-fv 6571 df-ov 7434 |
This theorem is referenced by: unirnioo 13486 ioorebas 13488 yonffthlem 18339 gsumval2a 18711 efginvrel2 19760 efgredleme 19776 efgcpbllemb 19788 mplsubrglem 22042 lecldbas 23243 blelrnps 24442 blelrn 24443 blssioo 24831 tgioo 24832 opnmbllem 25650 mbfdm 25675 mbfima 25679 tpr2rico 33873 dya2icoseg 34259 opnmbllem0 37643 elrnmpoid 45171 smflimlem3 46729 |
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