![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 6650 | . 2 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵)) | |
5 | 1, 2, 3, 4 | syl21anc 836 | 1 ⊢ (𝜑 → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∪ cun 3942 ∩ cin 3943 ∅c0 4318 Fn wfn 6527 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-fun 6534 df-fn 6535 |
This theorem is referenced by: fnunop 6652 brwdom2 9550 sseqfn 33220 bnj927 33611 metakunt19 40808 metakunt25 40814 ofun 40870 tfsconcatfn 41859 |
Copyright terms: Public domain | W3C validator |