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| Description: Weak version of funex 7239 that holds without ax-rep 5279. 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 7770 | . . 3 ⊢ ((dom 𝐹 ∈ 𝐵 ∧ ran 𝐹 ∈ 𝐶) → (dom 𝐹 × ran 𝐹) ∈ V) | |
| 2 | 1 | 3adant1 1131 | . 2 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ 𝐵 ∧ ran 𝐹 ∈ 𝐶) → (dom 𝐹 × ran 𝐹) ∈ V) | 
| 3 | funrel 6583 | . . . 4 ⊢ (Fun 𝐹 → Rel 𝐹) | |
| 4 | relssdmrn 6288 | . . . 4 ⊢ (Rel 𝐹 → 𝐹 ⊆ (dom 𝐹 × ran 𝐹)) | |
| 5 | 3, 4 | syl 17 | . . 3 ⊢ (Fun 𝐹 → 𝐹 ⊆ (dom 𝐹 × ran 𝐹)) | 
| 6 | 5 | 3ad2ant1 1134 | . 2 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ 𝐵 ∧ ran 𝐹 ∈ 𝐶) → 𝐹 ⊆ (dom 𝐹 × ran 𝐹)) | 
| 7 | 2, 6 | ssexd 5324 | 1 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ 𝐵 ∧ ran 𝐹 ∈ 𝐶) → 𝐹 ∈ V) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ w3a 1087 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 × cxp 5683 dom cdm 5685 ran crn 5686 Rel wrel 5690 Fun wfun 6555 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-fun 6563 | 
| This theorem is referenced by: mptexw 7977 mpoexw 8103 seqexw 14058 | 
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