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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 5421 | . . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → Fun 𝐹) | |
2 | funrnex 6093 | . . . 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 5420 | . . . . 5 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵) | |
5 | fdm 5353 | . . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴) | |
6 | 4, 5 | syl 14 | . . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → dom 𝐹 = 𝐴) |
7 | 6 | eleq1d 2239 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → (dom 𝐹 ∈ 𝐶 ↔ 𝐴 ∈ 𝐶)) |
8 | forn 5423 | . . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵) | |
9 | 8 | eleq1d 2239 | . . 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 1348 ∈ wcel 2141 Vcvv 2730 dom cdm 4611 ran crn 4612 Fun wfun 5192 ⟶wf 5194 –onto→wfo 5196 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 |
This theorem is referenced by: f1dmex 6095 f1oeng 6735 ctfoex 7095 omctfn 12398 |
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