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Mirrors > Home > MPE Home > Th. List > mptexw | Structured version Visualization version GIF version |
Description: Weak version of mptex 7235 that holds without ax-rep 5285. If the domain and codomain of a function given by maps-to notation are sets, the function is a set. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
Ref | Expression |
---|---|
mptexw.1 | ⊢ 𝐴 ∈ V |
mptexw.2 | ⊢ 𝐶 ∈ V |
mptexw.3 | ⊢ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 |
Ref | Expression |
---|---|
mptexw | ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funmpt 6591 | . 2 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵) | |
2 | mptexw.1 | . . 3 ⊢ 𝐴 ∈ V | |
3 | eqid 2728 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
4 | 3 | dmmptss 6245 | . . 3 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝐴 |
5 | 2, 4 | ssexi 5322 | . 2 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V |
6 | mptexw.2 | . . 3 ⊢ 𝐶 ∈ V | |
7 | mptexw.3 | . . . 4 ⊢ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 | |
8 | 3 | rnmptss 7133 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝐶) |
9 | 7, 8 | ax-mp 5 | . . 3 ⊢ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝐶 |
10 | 6, 9 | ssexi 5322 | . 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V |
11 | funexw 7955 | . 2 ⊢ ((Fun (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V ∧ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) | |
12 | 1, 5, 10, 11 | mp3an 1458 | 1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2099 ∀wral 3058 Vcvv 3471 ⊆ wss 3947 ↦ cmpt 5231 dom cdm 5678 ran crn 5679 Fun wfun 6542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-fun 6550 df-fn 6551 df-f 6552 |
This theorem is referenced by: grpinvfval 18935 odfval 19487 |
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