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Mirrors > Home > MPE Home > Th. List > fornex | Structured version Visualization version 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 6584 | . . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → Fun 𝐹) | |
2 | funrnex 7644 | . . . 4 ⊢ (dom 𝐹 ∈ 𝐶 → (Fun 𝐹 → ran 𝐹 ∈ V)) | |
3 | 1, 2 | syl5com 31 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → (dom 𝐹 ∈ 𝐶 → ran 𝐹 ∈ V)) |
4 | fof 6583 | . . . . 5 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵) | |
5 | 4 | fdmd 6516 | . . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → dom 𝐹 = 𝐴) |
6 | 5 | eleq1d 2894 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → (dom 𝐹 ∈ 𝐶 ↔ 𝐴 ∈ 𝐶)) |
7 | forn 6586 | . . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵) | |
8 | 7 | eleq1d 2894 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → (ran 𝐹 ∈ V ↔ 𝐵 ∈ V)) |
9 | 3, 6, 8 | 3imtr3d 294 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐴 ∈ 𝐶 → 𝐵 ∈ V)) |
10 | 9 | com12 32 | 1 ⊢ (𝐴 ∈ 𝐶 → (𝐹:𝐴–onto→𝐵 → 𝐵 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 Vcvv 3492 dom cdm 5548 ran crn 5549 Fun wfun 6342 –onto→wfo 6346 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 |
This theorem is referenced by: f1dmex 7647 f1ovv 7648 f1oeng 8516 fodomnum 9471 ttukeylem1 9919 fodomb 9936 cnexALT 12373 imasbas 16773 imasds 16774 elqtop 22233 qtoprest 22253 indishmph 22334 imasf1oxmet 22912 foresf1o 30192 noprc 33146 sge0f1o 42541 sge0fodjrnlem 42575 |
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