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| 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 520 | . . 3 ⊢ (𝜑 → ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐵)) |
| 6 | 1, 2, 5 | 3jca 1144 | . 2 ⊢ (𝜑 → (𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐵))) |
| 7 | fvun2 6973 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐵)) → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐺‘𝑋)) | |
| 8 | 6, 7 | syl 18 | 1 ⊢ (𝜑 → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐺‘𝑋)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 400 ∧ w3a 1101 = wceq 1568 ∈ wcel 2141 ∪ cun 3902 ∩ cin 3903 ∅c0 4285 Fn wfn 6531 ‘cfv 6536 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2143 ax-9 2151 ax-10 2174 ax-12 2211 ax-ext 2733 ax-sep 5256 ax-nul 5268 ax-pr 5404 |
| This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2095 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-ne 2957 df-ral 3078 df-rex 3088 df-rab 3415 df-v 3455 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4487 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-br 5109 df-opab 5173 df-id 5556 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-iota 6492 df-fun 6538 df-fn 6539 df-fv 6544 |
| This theorem is referenced by: noetainflem4 27880 ofun 42974 tfsconcatfv2 44037 |
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