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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 511 | . . 3 ⊢ (𝜑 → ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐵)) |
6 | 1, 2, 5 | 3jca 1127 | . 2 ⊢ (𝜑 → (𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐵))) |
7 | fvun2 7000 | . 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 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ∪ cun 3960 ∩ cin 3961 ∅c0 4338 Fn wfn 6557 ‘cfv 6562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-fv 6570 |
This theorem is referenced by: noetainflem4 27799 metakunt20 42205 metakunt22 42207 ofun 42255 tfsconcatfv2 43329 |
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