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Mirrors > Home > ILE Home > Th. List > focdmex | GIF version |
Description: If the domain of an onto function exists, so does its codomain. (Contributed by NM, 23-Jul-2004.) |
Ref | Expression |
---|---|
focdmex | ⊢ (𝐴 ∈ 𝐶 → (𝐹:𝐴–onto→𝐵 → 𝐵 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fofun 5455 | . . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → Fun 𝐹) | |
2 | funrnex 6134 | . . . 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 5454 | . . . . 5 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵) | |
5 | fdm 5387 | . . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴) | |
6 | 4, 5 | syl 14 | . . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → dom 𝐹 = 𝐴) |
7 | 6 | eleq1d 2258 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → (dom 𝐹 ∈ 𝐶 ↔ 𝐴 ∈ 𝐶)) |
8 | forn 5457 | . . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵) | |
9 | 8 | eleq1d 2258 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → (ran 𝐹 ∈ V ↔ 𝐵 ∈ V)) |
10 | 3, 7, 9 | 3imtr3d 202 | . 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 1364 ∈ wcel 2160 Vcvv 2752 dom cdm 4641 ran crn 4642 Fun wfun 5226 ⟶wf 5228 –onto→wfo 5230 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5234 df-fn 5235 df-f 5236 df-f1 5237 df-fo 5238 df-f1o 5239 df-fv 5240 |
This theorem is referenced by: f1dmex 6136 f1oeng 6778 ctfoex 7142 ennnfonelemj0 12447 ennnfonelemg 12449 omctfn 12489 imasival 12776 imasbas 12777 imasplusg 12778 |
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