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Mirrors > Home > MPE Home > Th. List > fnund | Structured version Visualization version GIF version |
Description: The union of two functions with disjoint domains, a deduction version. (Contributed by metakunt, 28-May-2024.) |
Ref | Expression |
---|---|
fnund.1 | ⊢ (𝜑 → 𝐹 Fn 𝐴) |
fnund.2 | ⊢ (𝜑 → 𝐺 Fn 𝐵) |
fnund.3 | ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅) |
Ref | Expression |
---|---|
fnund | ⊢ (𝜑 → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnund.1 | . 2 ⊢ (𝜑 → 𝐹 Fn 𝐴) | |
2 | fnund.2 | . 2 ⊢ (𝜑 → 𝐺 Fn 𝐵) | |
3 | fnund.3 | . 2 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅) | |
4 | fnun 6438 | . 2 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵)) | |
5 | 1, 2, 3, 4 | syl21anc 837 | 1 ⊢ (𝜑 → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∪ cun 3852 ∩ cin 3853 ∅c0 4221 Fn wfn 6323 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5162 ax-nul 5169 ax-pr 5291 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3an 1087 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ral 3073 df-rab 3077 df-v 3409 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-nul 4222 df-if 4414 df-sn 4516 df-pr 4518 df-op 4522 df-br 5026 df-opab 5088 df-id 5423 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-fun 6330 df-fn 6331 |
This theorem is referenced by: fnunop 6440 brwdom2 9055 sseqfn 31861 bnj927 32253 metakunt19 39650 metakunt25 39656 ofun 39702 |
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