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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 5441 | . . 3 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → 〈𝐶, 𝐷〉 ∈ (𝐴 × 𝐵)) | |
2 | df-ov 6977 | . . . 4 ⊢ (𝐶𝐹𝐷) = (𝐹‘〈𝐶, 𝐷〉) | |
3 | fnfvelrn 6671 | . . . 4 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 〈𝐶, 𝐷〉 ∈ (𝐴 × 𝐵)) → (𝐹‘〈𝐶, 𝐷〉) ∈ ran 𝐹) | |
4 | 2, 3 | syl5eqel 2867 | . . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 〈𝐶, 𝐷〉 ∈ (𝐴 × 𝐵)) → (𝐶𝐹𝐷) ∈ ran 𝐹) |
5 | 1, 4 | sylan2 583 | . 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → (𝐶𝐹𝐷) ∈ ran 𝐹) |
6 | 5 | 3impb 1095 | 1 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (𝐶𝐹𝐷) ∈ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∧ w3a 1068 ∈ wcel 2048 〈cop 4445 × cxp 5402 ran crn 5405 Fn wfn 6181 ‘cfv 6186 (class class class)co 6974 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2747 ax-sep 5058 ax-nul 5065 ax-pr 5184 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2756 df-cleq 2768 df-clel 2843 df-nfc 2915 df-ral 3090 df-rex 3091 df-rab 3094 df-v 3414 df-sbc 3681 df-dif 3831 df-un 3833 df-in 3835 df-ss 3842 df-nul 4178 df-if 4349 df-sn 4440 df-pr 4442 df-op 4446 df-uni 4711 df-br 4928 df-opab 4990 df-id 5309 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-iota 6150 df-fun 6188 df-fn 6189 df-fv 6194 df-ov 6977 |
This theorem is referenced by: unirnioo 12650 ioorebas 12652 yonffthlem 17384 gsumval2a 17741 efginvrel2 18605 efgredleme 18622 efgcpbllemb 18635 mplsubrglem 19927 lecldbas 21525 blelrnps 22723 blelrn 22724 blssioo 23100 tgioo 23101 opnmbllem 23899 mbfdm 23924 mbfima 23928 tpr2rico 30790 dya2icoseg 31171 opnmbllem0 34347 elrnmpt2id 40899 smflimlem3 42459 |
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