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Mirrors > Home > MPE Home > Th. List > funexw | Structured version Visualization version GIF version |
Description: Weak version of funex 7170 that holds without ax-rep 5243. 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 7685 | . . 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 6519 | . . . 4 ⊢ (Fun 𝐹 → Rel 𝐹) | |
4 | relssdmrn 6221 | . . . 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 5282 | 1 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ 𝐵 ∧ ran 𝐹 ∈ 𝐶) → 𝐹 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1088 ∈ wcel 2107 Vcvv 3444 ⊆ wss 3911 × cxp 5632 dom cdm 5634 ran crn 5635 Rel wrel 5639 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-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-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 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-xp 5640 df-rel 5641 df-cnv 5642 df-dm 5644 df-rn 5645 df-fun 6499 |
This theorem is referenced by: mptexw 7886 mpoexw 8012 seqexw 13928 |
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