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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 6802 | . . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵) | |
2 | 1 | eleq2d 2813 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → (𝑌 ∈ ran 𝐹 ↔ 𝑌 ∈ 𝐵)) |
3 | fofn 6801 | . . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴) | |
4 | fvelrnb 6946 | . . . 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 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∃wrex 3064 ran crn 5670 Fn wfn 6532 –onto→wfo 6535 ‘cfv 6537 |
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-sep 5292 ax-nul 5299 ax-pr 5420 |
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-rab 3427 df-v 3470 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-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-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-fo 6543 df-fv 6545 |
This theorem is referenced by: mhmid 18991 mhmmnd 18992 ghmgrp 18994 symgmov2 19307 ghmcmn 19751 imasabl 19796 founiiun 44450 founiiun0 44461 sge0f1o 45670 isomenndlem 45818 ovnsubaddlem1 45858 f1oresf1o2 46571 |
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