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Mirrors > Home > MPE Home > Th. List > dff4 | Structured version Visualization version GIF version |
Description: Alternate definition of a mapping. (Contributed by NM, 20-Mar-2007.) |
Ref | Expression |
---|---|
dff4 | ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥𝐹𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dff3 6878 | . 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦)) | |
2 | df-br 5031 | . . . . . . . 8 ⊢ (𝑥𝐹𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐹) | |
3 | ssel 3870 | . . . . . . . . 9 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (〈𝑥, 𝑦〉 ∈ 𝐹 → 〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵))) | |
4 | opelxp2 5567 | . . . . . . . . 9 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) → 𝑦 ∈ 𝐵) | |
5 | 3, 4 | syl6 35 | . . . . . . . 8 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (〈𝑥, 𝑦〉 ∈ 𝐹 → 𝑦 ∈ 𝐵)) |
6 | 2, 5 | syl5bi 245 | . . . . . . 7 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (𝑥𝐹𝑦 → 𝑦 ∈ 𝐵)) |
7 | 6 | pm4.71rd 566 | . . . . . 6 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (𝑥𝐹𝑦 ↔ (𝑦 ∈ 𝐵 ∧ 𝑥𝐹𝑦))) |
8 | 7 | eubidv 2587 | . . . . 5 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (∃!𝑦 𝑥𝐹𝑦 ↔ ∃!𝑦(𝑦 ∈ 𝐵 ∧ 𝑥𝐹𝑦))) |
9 | df-reu 3060 | . . . . 5 ⊢ (∃!𝑦 ∈ 𝐵 𝑥𝐹𝑦 ↔ ∃!𝑦(𝑦 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) | |
10 | 8, 9 | bitr4di 292 | . . . 4 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (∃!𝑦 𝑥𝐹𝑦 ↔ ∃!𝑦 ∈ 𝐵 𝑥𝐹𝑦)) |
11 | 10 | ralbidv 3109 | . . 3 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → (∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦 ↔ ∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥𝐹𝑦)) |
12 | 11 | pm5.32i 578 | . 2 ⊢ ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦) ↔ (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥𝐹𝑦)) |
13 | 1, 12 | bitri 278 | 1 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥𝐹𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∈ wcel 2114 ∃!weu 2569 ∀wral 3053 ∃!wreu 3055 ⊆ wss 3843 〈cop 4522 class class class wbr 5030 × cxp 5523 ⟶wf 6335 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-v 3400 df-sbc 3681 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-fv 6347 |
This theorem is referenced by: exfo 6883 |
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