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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 6805 | . . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵) | |
2 | 1 | eleq2d 2819 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → (𝑌 ∈ ran 𝐹 ↔ 𝑌 ∈ 𝐵)) |
3 | fofn 6804 | . . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴) | |
4 | fvelrnb 6949 | . . . 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 280 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝑌 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑌)) |
7 | 6 | biimpa 477 | 1 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑌 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∃wrex 3070 ran crn 5676 Fn wfn 6535 –onto→wfo 6538 ‘cfv 6540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-fo 6546 df-fv 6548 |
This theorem is referenced by: mhmid 18940 mhmmnd 18941 ghmgrp 18943 symgmov2 19249 ghmcmn 19693 imasabl 19738 founiiun 43860 founiiun0 43873 sge0f1o 45084 isomenndlem 45232 ovnsubaddlem1 45272 f1oresf1o2 45985 |
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