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Mirrors > Home > MPE Home > Th. List > focdmex | 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 |
---|---|
focdmex | ⊢ (𝐴 ∈ 𝐶 → (𝐹:𝐴–onto→𝐵 → 𝐵 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fofun 6799 | . . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → Fun 𝐹) | |
2 | funrnex 7936 | . . . 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 6798 | . . . . 5 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵) | |
5 | 4 | fdmd 6721 | . . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → dom 𝐹 = 𝐴) |
6 | 5 | eleq1d 2812 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → (dom 𝐹 ∈ 𝐶 ↔ 𝐴 ∈ 𝐶)) |
7 | forn 6801 | . . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵) | |
8 | 7 | eleq1d 2812 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → (ran 𝐹 ∈ V ↔ 𝐵 ∈ V)) |
9 | 3, 6, 8 | 3imtr3d 293 | . 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 2098 Vcvv 3468 dom cdm 5669 ran crn 5670 Fun wfun 6530 –onto→wfo 6534 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 |
This theorem is referenced by: f1dmex 7939 f1ovv 7940 fsetprcnex 8855 f1oeng 8966 fodomnum 10051 ttukeylem1 10503 fodomb 10520 cnexALT 12971 hasheqf1oi 14313 imasbas 17464 imasds 17465 elqtop 23551 qtoprest 23571 indishmph 23652 imasf1oxmet 24231 noprc 27662 foresf1o 32246 sge0f1o 45652 sge0fodjrnlem 45686 |
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