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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 7132 | . 2 ⊢ Fun 𝐹 |
3 | 1 | dmmpossx 7620 | . . 3 ⊢ dom 𝐹 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) |
4 | snex 5223 | . . . . . 6 ⊢ {𝑥} ∈ V | |
5 | xpexg 7330 | . . . . . 6 ⊢ (({𝑥} ∈ V ∧ 𝐵 ∈ 𝑆) → ({𝑥} × 𝐵) ∈ V) | |
6 | 4, 5 | mpan 686 | . . . . 5 ⊢ (𝐵 ∈ 𝑆 → ({𝑥} × 𝐵) ∈ V) |
7 | 6 | ralimi 3127 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 → ∀𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∈ V) |
8 | iunexg 7520 | . . . 4 ⊢ ((𝐴 ∈ 𝑅 ∧ ∀𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∈ V) → ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∈ V) | |
9 | 7, 8 | sylan2 592 | . . 3 ⊢ ((𝐴 ∈ 𝑅 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆) → ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∈ V) |
10 | ssexg 5118 | . . 3 ⊢ ((dom 𝐹 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∧ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∈ V) → dom 𝐹 ∈ V) | |
11 | 3, 9, 10 | sylancr 587 | . 2 ⊢ ((𝐴 ∈ 𝑅 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆) → dom 𝐹 ∈ V) |
12 | funex 6848 | . 2 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ V) → 𝐹 ∈ V) | |
13 | 2, 11, 12 | sylancr 587 | 1 ⊢ ((𝐴 ∈ 𝑅 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆) → 𝐹 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1522 ∈ wcel 2081 ∀wral 3105 Vcvv 3437 ⊆ wss 3859 {csn 4472 ∪ ciun 4825 × cxp 5441 dom cdm 5443 Fun wfun 6219 ∈ cmpo 7018 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-id 5348 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-oprab 7020 df-mpo 7021 df-1st 7545 df-2nd 7546 |
This theorem is referenced by: mpoexg 7630 mpoex 7633 gsum2d2lem 18813 taylfval 24630 ptrest 34422 |
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