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| Mirrors > Home > MPE Home > Th. List > mpoexw | Structured version Visualization version GIF version | ||
| Description: Weak version of mpoex 8105 that holds without ax-rep 5278. If the domain and codomain of an operation given by maps-to notation are sets, the operation is a set. (Contributed by Rohan Ridenour, 14-Aug-2023.) | 
| Ref | Expression | 
|---|---|
| mpoexw.1 | ⊢ 𝐴 ∈ V | 
| mpoexw.2 | ⊢ 𝐵 ∈ V | 
| mpoexw.3 | ⊢ 𝐷 ∈ V | 
| mpoexw.4 | ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 | 
| Ref | Expression | 
|---|---|
| mpoexw | ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqid 2736 | . . 3 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) | |
| 2 | 1 | mpofun 7558 | . 2 ⊢ Fun (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) | 
| 3 | mpoexw.4 | . . . 4 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 | |
| 4 | 1 | dmmpoga 8099 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 → dom (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝐴 × 𝐵)) | 
| 5 | 3, 4 | ax-mp 5 | . . 3 ⊢ dom (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝐴 × 𝐵) | 
| 6 | mpoexw.1 | . . . 4 ⊢ 𝐴 ∈ V | |
| 7 | mpoexw.2 | . . . 4 ⊢ 𝐵 ∈ V | |
| 8 | 6, 7 | xpex 7774 | . . 3 ⊢ (𝐴 × 𝐵) ∈ V | 
| 9 | 5, 8 | eqeltri 2836 | . 2 ⊢ dom (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V | 
| 10 | 1 | rnmpo 7567 | . . 3 ⊢ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶} | 
| 11 | mpoexw.3 | . . . 4 ⊢ 𝐷 ∈ V | |
| 12 | 3 | rspec 3249 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷) | 
| 13 | 12 | r19.21bi 3250 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝐷) | 
| 14 | eleq1a 2835 | . . . . . . . 8 ⊢ (𝐶 ∈ 𝐷 → (𝑧 = 𝐶 → 𝑧 ∈ 𝐷)) | |
| 15 | 13, 14 | syl 17 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑧 = 𝐶 → 𝑧 ∈ 𝐷)) | 
| 16 | 15 | rexlimdva 3154 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝑧 ∈ 𝐷)) | 
| 17 | 16 | rexlimiv 3147 | . . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝑧 ∈ 𝐷) | 
| 18 | 17 | abssi 4069 | . . . 4 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶} ⊆ 𝐷 | 
| 19 | 11, 18 | ssexi 5321 | . . 3 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶} ∈ V | 
| 20 | 10, 19 | eqeltri 2836 | . 2 ⊢ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V | 
| 21 | funexw 7977 | . 2 ⊢ ((Fun (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∧ dom (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V ∧ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V) | |
| 22 | 2, 9, 20, 21 | mp3an 1462 | 1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 {cab 2713 ∀wral 3060 ∃wrex 3069 Vcvv 3479 × cxp 5682 dom cdm 5684 ran crn 5685 Fun wfun 6554 ∈ cmpo 7434 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fv 6568 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016 | 
| This theorem is referenced by: mptmpoopabbrd 8106 prdsvallem 17500 prdsds 17510 plusffval 18660 grpsubfval 19002 mulgfval 19088 scaffval 20879 ipffval 21667 | 
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