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Mirrors > Home > MPE Home > Th. List > mptexw | Structured version Visualization version GIF version |
Description: Weak version of mptex 7243 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 6606 | . 2 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵) | |
2 | mptexw.1 | . . 3 ⊢ 𝐴 ∈ V | |
3 | eqid 2735 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
4 | 3 | dmmptss 6263 | . . 3 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝐴 |
5 | 2, 4 | ssexi 5328 | . 2 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V |
6 | mptexw.2 | . . 3 ⊢ 𝐶 ∈ V | |
7 | mptexw.3 | . . . 4 ⊢ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 | |
8 | 3 | rnmptss 7143 | . . . 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 5328 | . 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V |
11 | funexw 7975 | . 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 1460 | 1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 ∀wral 3059 Vcvv 3478 ⊆ wss 3963 ↦ cmpt 5231 dom cdm 5689 ran crn 5690 Fun wfun 6557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-fun 6565 df-fn 6566 df-f 6567 |
This theorem is referenced by: grpinvfval 19009 odfval 19565 |
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