Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > fvun1d | Structured version Visualization version GIF version |
Description: The value of a union when the argument is in the first domain, a deduction version. (Contributed by metakunt, 28-May-2024.) |
Ref | Expression |
---|---|
fvun1d.1 | ⊢ (𝜑 → 𝐹 Fn 𝐴) |
fvun1d.2 | ⊢ (𝜑 → 𝐺 Fn 𝐵) |
fvun1d.3 | ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅) |
fvun1d.4 | ⊢ (𝜑 → 𝑋 ∈ 𝐴) |
Ref | Expression |
---|---|
fvun1d | ⊢ (𝜑 → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐹‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvun1d.1 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴) | |
2 | fvun1d.2 | . . 3 ⊢ (𝜑 → 𝐺 Fn 𝐵) | |
3 | fvun1d.3 | . . . 4 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅) | |
4 | fvun1d.4 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
5 | 3, 4 | jca 515 | . . 3 ⊢ (𝜑 → ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) |
6 | 1, 2, 5 | 3jca 1130 | . 2 ⊢ (𝜑 → (𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴))) |
7 | fvun1 6780 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐹‘𝑋)) | |
8 | 6, 7 | syl 17 | 1 ⊢ (𝜑 → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐹‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ∪ cun 3851 ∩ cin 3852 ∅c0 4223 Fn wfn 6353 ‘cfv 6358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-fv 6366 |
This theorem is referenced by: hashf1lem1 13985 hashf1lem1OLD 13986 elrspunidl 31274 metakunt21 39808 metakunt22 39809 ofun 39865 |
Copyright terms: Public domain | W3C validator |