Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1422 | Structured version Visualization version GIF version |
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj1422.1 | ⊢ (𝜑 → Fun 𝐴) |
bnj1422.2 | ⊢ (𝜑 → dom 𝐴 = 𝐵) |
Ref | Expression |
---|---|
bnj1422 | ⊢ (𝜑 → 𝐴 Fn 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1422.1 | . 2 ⊢ (𝜑 → Fun 𝐴) | |
2 | bnj1422.2 | . 2 ⊢ (𝜑 → dom 𝐴 = 𝐵) | |
3 | df-fn 6383 | . 2 ⊢ (𝐴 Fn 𝐵 ↔ (Fun 𝐴 ∧ dom 𝐴 = 𝐵)) | |
4 | 1, 2, 3 | sylanbrc 586 | 1 ⊢ (𝜑 → 𝐴 Fn 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 dom cdm 5551 Fun wfun 6374 Fn wfn 6375 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 210 df-an 400 df-fn 6383 |
This theorem is referenced by: bnj150 32569 bnj535 32583 bnj1312 32751 bnj60 32755 |
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