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Mirrors > Home > MPE Home > Th. List > foelcdmi | Structured version Visualization version GIF version |
Description: A member of a surjective function's codomain is a value of the function. (Contributed by Thierry Arnoux, 23-Jan-2020.) |
Ref | Expression |
---|---|
foelcdmi | ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑌 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | forn 6824 | . . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵) | |
2 | 1 | eleq2d 2825 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → (𝑌 ∈ ran 𝐹 ↔ 𝑌 ∈ 𝐵)) |
3 | fofn 6823 | . . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴) | |
4 | fvelrnb 6969 | . . . 4 ⊢ (𝐹 Fn 𝐴 → (𝑌 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑌)) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → (𝑌 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑌)) |
6 | 2, 5 | bitr3d 281 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝑌 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑌)) |
7 | 6 | biimpa 476 | 1 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑌 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 ran crn 5690 Fn wfn 6558 –onto→wfo 6561 ‘cfv 6563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fo 6569 df-fv 6571 |
This theorem is referenced by: mhmid 19094 mhmmnd 19095 ghmgrp 19097 symgmov2 19420 ghmcmn 19864 imasabl 19909 mndlactfo 33015 mndractfo 33017 founiiun 45122 founiiun0 45133 sge0f1o 46338 isomenndlem 46486 ovnsubaddlem1 46526 f1oresf1o2 47241 grimuhgr 47816 grimcnv 47817 |
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