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Mirrors > Home > ILE Home > Th. List > dff4im | GIF version |
Description: Property of a mapping. (Contributed by Jim Kingdon, 4-Jan-2019.) |
Ref | Expression |
---|---|
dff4im | ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥𝐹𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dff3im 5660 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦)) | |
2 | df-br 4003 | . . . . . . . 8 ⊢ (𝑥𝐹𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐹) | |
3 | ssel 3149 | . . . . . . . . 9 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (〈𝑥, 𝑦〉 ∈ 𝐹 → 〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵))) | |
4 | opelxp2 4660 | . . . . . . . . 9 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) → 𝑦 ∈ 𝐵) | |
5 | 3, 4 | syl6 33 | . . . . . . . 8 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (〈𝑥, 𝑦〉 ∈ 𝐹 → 𝑦 ∈ 𝐵)) |
6 | 2, 5 | biimtrid 152 | . . . . . . 7 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (𝑥𝐹𝑦 → 𝑦 ∈ 𝐵)) |
7 | 6 | pm4.71rd 394 | . . . . . 6 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (𝑥𝐹𝑦 ↔ (𝑦 ∈ 𝐵 ∧ 𝑥𝐹𝑦))) |
8 | 7 | eubidv 2034 | . . . . 5 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (∃!𝑦 𝑥𝐹𝑦 ↔ ∃!𝑦(𝑦 ∈ 𝐵 ∧ 𝑥𝐹𝑦))) |
9 | df-reu 2462 | . . . . 5 ⊢ (∃!𝑦 ∈ 𝐵 𝑥𝐹𝑦 ↔ ∃!𝑦(𝑦 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) | |
10 | 8, 9 | bitr4di 198 | . . . 4 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (∃!𝑦 𝑥𝐹𝑦 ↔ ∃!𝑦 ∈ 𝐵 𝑥𝐹𝑦)) |
11 | 10 | ralbidv 2477 | . . 3 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦 ↔ ∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥𝐹𝑦)) |
12 | 11 | pm5.32i 454 | . 2 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦) ↔ (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥𝐹𝑦)) |
13 | 1, 12 | sylib 122 | 1 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥𝐹𝑦)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∃!weu 2026 ∈ wcel 2148 ∀wral 2455 ∃!wreu 2457 ⊆ wss 3129 〈cop 3595 class class class wbr 4002 × cxp 4623 ⟶wf 5211 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 ax-pr 4208 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-reu 2462 df-v 2739 df-sbc 2963 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4003 df-opab 4064 df-id 4292 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-iota 5177 df-fun 5217 df-fn 5218 df-f 5219 df-fv 5223 |
This theorem is referenced by: (None) |
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