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| Mirrors > Home > MPE Home > Th. List > mpoexxg | Structured version Visualization version GIF version | ||
| Description: Existence of an operation class abstraction (version for dependent domains). (Contributed by Mario Carneiro, 30-Dec-2016.) |
| Ref | Expression |
|---|---|
| mpoexg.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) |
| Ref | Expression |
|---|---|
| mpoexxg | ⊢ ((𝐴 ∈ 𝑅 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆) → 𝐹 ∈ V) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mpoexg.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) | |
| 2 | 1 | mpofun 7540 | . 2 ⊢ Fun 𝐹 |
| 3 | 1 | dmmpossx 8065 | . . 3 ⊢ dom 𝐹 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) |
| 4 | vsnex 5409 | . . . . . 6 ⊢ {𝑥} ∈ V | |
| 5 | xpexg 7751 | . . . . . 6 ⊢ (({𝑥} ∈ V ∧ 𝐵 ∈ 𝑆) → ({𝑥} × 𝐵) ∈ V) | |
| 6 | 4, 5 | mpan 703 | . . . . 5 ⊢ (𝐵 ∈ 𝑆 → ({𝑥} × 𝐵) ∈ V) |
| 7 | 6 | ralimi 3105 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 → ∀𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∈ V) |
| 8 | iunexg 7962 | . . . 4 ⊢ ((𝐴 ∈ 𝑅 ∧ ∀𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∈ V) → ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∈ V) | |
| 9 | 7, 8 | sylan2 605 | . . 3 ⊢ ((𝐴 ∈ 𝑅 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆) → ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∈ V) |
| 10 | ssexg 5293 | . . 3 ⊢ ((dom 𝐹 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∧ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∈ V) → dom 𝐹 ∈ V) | |
| 11 | 3, 9, 10 | sylancr 599 | . 2 ⊢ ((𝐴 ∈ 𝑅 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆) → dom 𝐹 ∈ V) |
| 12 | funex 7221 | . 2 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ V) → 𝐹 ∈ V) | |
| 13 | 2, 11, 12 | sylancr 599 | 1 ⊢ ((𝐴 ∈ 𝑅 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆) → 𝐹 ∈ V) |
| Colors of variables: wff setvar class |
| This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ∀wral 3082 Vcvv 3458 ⊆ wss 3908 {csn 4592 ∪ ciun 4959 × cxp 5662 dom cdm 5664 Fun wfun 6534 ∈ cmpo 7418 |
| This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5241 ax-sep 5260 ax-nul 5272 ax-pow 5339 ax-pr 5407 ax-un 7738 |
| This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-ral 3083 df-rex 3093 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4290 df-if 4491 df-pw 4567 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4876 df-iun 4961 df-br 5113 df-opab 5177 df-mpt 5196 df-id 5559 df-xp 5670 df-rel 5671 df-cnv 5672 df-co 5673 df-dm 5674 df-rn 5675 df-res 5676 df-ima 5677 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-oprab 7420 df-mpo 7421 df-1st 7988 df-2nd 7989 |
| This theorem is used by: mpoexg 8075 mpoex 8078 mpoexd 8079 gsum2d2lem 20053 taylfval 26537 ptrest 38302 |
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