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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 5350). (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 5350 | . . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ 𝐴) = 𝐵) | |
2 | 1 | eqcomd 2145 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐵 = (𝐹 “ 𝐴)) |
3 | 2 | eleq2d 2209 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝑌 ∈ 𝐵 ↔ 𝑌 ∈ (𝐹 “ 𝐴))) |
4 | fofn 5347 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴) | |
5 | ssid 3117 | . . 3 ⊢ 𝐴 ⊆ 𝐴 | |
6 | fvelimab 5477 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ⊆ 𝐴) → (𝑌 ∈ (𝐹 “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑌)) | |
7 | eqcom 2141 | . . . . 5 ⊢ ((𝐹‘𝑥) = 𝑌 ↔ 𝑌 = (𝐹‘𝑥)) | |
8 | 7 | rexbii 2442 | . . . 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 409 | . 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 1331 ∈ wcel 1480 ∃wrex 2417 ⊆ wss 3071 “ cima 4542 Fn wfn 5118 –onto→wfo 5121 ‘cfv 5123 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fo 5129 df-fv 5131 |
This theorem is referenced by: foelrn 5654 |
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