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Mirrors > Home > ILE Home > Th. List > fornex | GIF version |
Description: If the domain of an onto function exists, so does its codomain. (Contributed by NM, 23-Jul-2004.) |
Ref | Expression |
---|---|
fornex | ⊢ (𝐴 ∈ 𝐶 → (𝐹:𝐴–onto→𝐵 → 𝐵 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fofun 5346 | . . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → Fun 𝐹) | |
2 | funrnex 6012 | . . . 4 ⊢ (dom 𝐹 ∈ 𝐶 → (Fun 𝐹 → ran 𝐹 ∈ V)) | |
3 | 1, 2 | syl5com 29 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → (dom 𝐹 ∈ 𝐶 → ran 𝐹 ∈ V)) |
4 | fof 5345 | . . . . 5 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵) | |
5 | fdm 5278 | . . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴) | |
6 | 4, 5 | syl 14 | . . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → dom 𝐹 = 𝐴) |
7 | 6 | eleq1d 2208 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → (dom 𝐹 ∈ 𝐶 ↔ 𝐴 ∈ 𝐶)) |
8 | forn 5348 | . . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵) | |
9 | 8 | eleq1d 2208 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → (ran 𝐹 ∈ V ↔ 𝐵 ∈ V)) |
10 | 3, 7, 9 | 3imtr3d 201 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐴 ∈ 𝐶 → 𝐵 ∈ V)) |
11 | 10 | com12 30 | 1 ⊢ (𝐴 ∈ 𝐶 → (𝐹:𝐴–onto→𝐵 → 𝐵 ∈ V)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 Vcvv 2686 dom cdm 4539 ran crn 4540 Fun wfun 5117 ⟶wf 5119 –onto→wfo 5121 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 |
This theorem is referenced by: f1dmex 6014 f1oeng 6651 ctfoex 7003 |
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