![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > mptexw | GIF version |
Description: Weak version of mptex 5755 that holds without ax-coll 4130. 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 5266 | . 2 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵) | |
2 | mptexw.1 | . . 3 ⊢ 𝐴 ∈ V | |
3 | eqid 2187 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
4 | 3 | dmmptss 5137 | . . 3 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝐴 |
5 | 2, 4 | ssexi 4153 | . 2 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V |
6 | mptexw.2 | . . 3 ⊢ 𝐶 ∈ V | |
7 | mptexw.3 | . . . 4 ⊢ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 | |
8 | 3 | rnmptss 5690 | . . . 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 4153 | . 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V |
11 | funexw 6126 | . 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 1347 | 1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2158 ∀wral 2465 Vcvv 2749 ⊆ wss 3141 ↦ cmpt 4076 dom cdm 4638 ran crn 4639 Fun wfun 5222 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-rab 2474 df-v 2751 df-sbc 2975 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-fv 5236 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |