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Mirrors > Home > MPE Home > Th. List > funexw | Structured version Visualization version GIF version |
Description: Weak version of funex 7205 that holds without ax-rep 5278. 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 7720 | . . 3 ⊢ ((dom 𝐹 ∈ 𝐵 ∧ ran 𝐹 ∈ 𝐶) → (dom 𝐹 × ran 𝐹) ∈ V) | |
2 | 1 | 3adant1 1130 | . 2 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ 𝐵 ∧ ran 𝐹 ∈ 𝐶) → (dom 𝐹 × ran 𝐹) ∈ V) |
3 | funrel 6554 | . . . 4 ⊢ (Fun 𝐹 → Rel 𝐹) | |
4 | relssdmrn 6256 | . . . 4 ⊢ (Rel 𝐹 → 𝐹 ⊆ (dom 𝐹 × ran 𝐹)) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (Fun 𝐹 → 𝐹 ⊆ (dom 𝐹 × ran 𝐹)) |
6 | 5 | 3ad2ant1 1133 | . 2 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ 𝐵 ∧ ran 𝐹 ∈ 𝐶) → 𝐹 ⊆ (dom 𝐹 × ran 𝐹)) |
7 | 2, 6 | ssexd 5317 | 1 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ 𝐵 ∧ ran 𝐹 ∈ 𝐶) → 𝐹 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1087 ∈ wcel 2106 Vcvv 3473 ⊆ wss 3944 × cxp 5667 dom cdm 5669 ran crn 5670 Rel wrel 5674 Fun wfun 6526 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-opab 5204 df-xp 5675 df-rel 5676 df-cnv 5677 df-dm 5679 df-rn 5680 df-fun 6534 |
This theorem is referenced by: mptexw 7921 mpoexw 8047 seqexw 13964 |
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