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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj519 | Structured version Visualization version GIF version |
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by Mario Carneiro, 6-May-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj519.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
bnj519 | ⊢ (𝐵 ∈ V → Fun {⟨𝐴, 𝐵⟩}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj519.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | funsng 6590 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → Fun {⟨𝐴, 𝐵⟩}) | |
3 | 1, 2 | mpan 687 | 1 ⊢ (𝐵 ∈ V → Fun {⟨𝐴, 𝐵⟩}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 Vcvv 3466 {csn 4621 ⟨cop 4627 Fun wfun 6528 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-mo 2526 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-br 5140 df-opab 5202 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-fun 6536 |
This theorem is referenced by: bnj97 34396 bnj535 34420 |
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