![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fvun2d | Structured version Visualization version GIF version |
Description: The value of a union when the argument is in the second domain, a deduction version. (Contributed by metakunt, 28-May-2024.) |
Ref | Expression |
---|---|
fvun2d.1 | ⊢ (𝜑 → 𝐹 Fn 𝐴) |
fvun2d.2 | ⊢ (𝜑 → 𝐺 Fn 𝐵) |
fvun2d.3 | ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅) |
fvun2d.4 | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
Ref | Expression |
---|---|
fvun2d | ⊢ (𝜑 → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐺‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvun2d.1 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴) | |
2 | fvun2d.2 | . . 3 ⊢ (𝜑 → 𝐺 Fn 𝐵) | |
3 | fvun2d.3 | . . . 4 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅) | |
4 | fvun2d.4 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
5 | 3, 4 | jca 516 | . . 3 ⊢ (𝜑 → ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐵)) |
6 | 1, 2, 5 | 3jca 1126 | . 2 ⊢ (𝜑 → (𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐵))) |
7 | fvun2 6737 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐵)) → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐺‘𝑋)) | |
8 | 6, 7 | syl 17 | 1 ⊢ (𝜑 → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐺‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 400 ∧ w3a 1085 = wceq 1539 ∈ wcel 2112 ∪ cun 3852 ∩ cin 3853 ∅c0 4221 Fn wfn 6323 ‘cfv 6328 |
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-ne 2950 df-ral 3073 df-rex 3074 df-rab 3077 df-v 3409 df-sbc 3694 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-uni 4792 df-br 5026 df-opab 5088 df-id 5423 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-iota 6287 df-fun 6330 df-fn 6331 df-fv 6336 |
This theorem is referenced by: noetainflem4 33493 metakunt20 39651 metakunt22 39653 ofun 39702 |
Copyright terms: Public domain | W3C validator |