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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 6604 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → Fun {⟨𝐴, 𝐵⟩}) | |
3 | 1, 2 | mpan 689 | 1 ⊢ (𝐵 ∈ V → Fun {⟨𝐴, 𝐵⟩}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 Vcvv 3471 {csn 4629 ⟨cop 4635 Fun wfun 6542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-mo 2530 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 df-opab 5211 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-fun 6550 |
This theorem is referenced by: bnj97 34497 bnj535 34521 |
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