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Mirrors > Home > MPE Home > Th. List > mptexw | Structured version Visualization version GIF version |
Description: Weak version of mptex 7174 that holds without ax-rep 5243. 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 6540 | . 2 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵) | |
2 | mptexw.1 | . . 3 ⊢ 𝐴 ∈ V | |
3 | eqid 2733 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
4 | 3 | dmmptss 6194 | . . 3 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝐴 |
5 | 2, 4 | ssexi 5280 | . 2 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V |
6 | mptexw.2 | . . 3 ⊢ 𝐶 ∈ V | |
7 | mptexw.3 | . . . 4 ⊢ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 | |
8 | 3 | rnmptss 7071 | . . . 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 5280 | . 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V |
11 | funexw 7885 | . 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 1462 | 1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 ∀wral 3061 Vcvv 3444 ⊆ wss 3911 ↦ cmpt 5189 dom cdm 5634 ran crn 5635 Fun wfun 6491 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-fun 6499 df-fn 6500 df-f 6501 |
This theorem is referenced by: grpinvfval 18794 odfval 19319 |
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