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Mirrors > Home > ILE Home > Th. List > foima2 | GIF version |
Description: Given an onto function, an element is in its codomain if and only if it is the image of an element of its domain (see foima 5358). (Contributed by BJ, 6-Jul-2022.) |
Ref | Expression |
---|---|
foima2 | ⊢ (𝐹:𝐴–onto→𝐵 → (𝑌 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑌 = (𝐹‘𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | foima 5358 | . . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ 𝐴) = 𝐵) | |
2 | 1 | eqcomd 2146 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐵 = (𝐹 “ 𝐴)) |
3 | 2 | eleq2d 2210 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝑌 ∈ 𝐵 ↔ 𝑌 ∈ (𝐹 “ 𝐴))) |
4 | fofn 5355 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴) | |
5 | ssid 3122 | . . 3 ⊢ 𝐴 ⊆ 𝐴 | |
6 | fvelimab 5485 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ⊆ 𝐴) → (𝑌 ∈ (𝐹 “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑌)) | |
7 | eqcom 2142 | . . . . 5 ⊢ ((𝐹‘𝑥) = 𝑌 ↔ 𝑌 = (𝐹‘𝑥)) | |
8 | 7 | rexbii 2445 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑌 ↔ ∃𝑥 ∈ 𝐴 𝑌 = (𝐹‘𝑥)) |
9 | 6, 8 | syl6bb 195 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ⊆ 𝐴) → (𝑌 ∈ (𝐹 “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝑌 = (𝐹‘𝑥))) |
10 | 4, 5, 9 | sylancl 410 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝑌 ∈ (𝐹 “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝑌 = (𝐹‘𝑥))) |
11 | 3, 10 | bitrd 187 | 1 ⊢ (𝐹:𝐴–onto→𝐵 → (𝑌 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑌 = (𝐹‘𝑥))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1332 ∈ wcel 1481 ∃wrex 2418 ⊆ wss 3076 “ cima 4550 Fn wfn 5126 –onto→wfo 5129 ‘cfv 5131 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-fo 5137 df-fv 5139 |
This theorem is referenced by: foelrn 5662 |
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