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Mirrors > Home > MPE Home > Th. List > funexw | Structured version Visualization version GIF version |
Description: Weak version of funex 7239 that holds without ax-rep 5285. If the domain and codomain of a function exist, so does the function. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
Ref | Expression |
---|---|
funexw | ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ 𝐵 ∧ ran 𝐹 ∈ 𝐶) → 𝐹 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpexg 7769 | . . 3 ⊢ ((dom 𝐹 ∈ 𝐵 ∧ ran 𝐹 ∈ 𝐶) → (dom 𝐹 × ran 𝐹) ∈ V) | |
2 | 1 | 3adant1 1129 | . 2 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ 𝐵 ∧ ran 𝐹 ∈ 𝐶) → (dom 𝐹 × ran 𝐹) ∈ V) |
3 | funrel 6585 | . . . 4 ⊢ (Fun 𝐹 → Rel 𝐹) | |
4 | relssdmrn 6290 | . . . 4 ⊢ (Rel 𝐹 → 𝐹 ⊆ (dom 𝐹 × ran 𝐹)) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (Fun 𝐹 → 𝐹 ⊆ (dom 𝐹 × ran 𝐹)) |
6 | 5 | 3ad2ant1 1132 | . 2 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ 𝐵 ∧ ran 𝐹 ∈ 𝐶) → 𝐹 ⊆ (dom 𝐹 × ran 𝐹)) |
7 | 2, 6 | ssexd 5330 | 1 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ 𝐵 ∧ ran 𝐹 ∈ 𝐶) → 𝐹 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 ∈ wcel 2106 Vcvv 3478 ⊆ wss 3963 × cxp 5687 dom cdm 5689 ran crn 5690 Rel wrel 5694 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-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-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 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-xp 5695 df-rel 5696 df-cnv 5697 df-dm 5699 df-rn 5700 df-fun 6565 |
This theorem is referenced by: mptexw 7976 mpoexw 8102 seqexw 14055 |
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