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Mirrors > Home > MPE Home > Th. List > funrnex | Structured version Visualization version GIF version |
Description: If the domain of a function exists, so does its range. Part of Theorem 4.15(v) of [Monk1] p. 46. This theorem is derived using the Axiom of Replacement in the form of funex 6959. (Contributed by NM, 11-Nov-1995.) |
Ref | Expression |
---|---|
funrnex | ⊢ (dom 𝐹 ∈ 𝐵 → (Fun 𝐹 → ran 𝐹 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funex 6959 | . . 3 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ 𝐵) → 𝐹 ∈ V) | |
2 | 1 | ex 416 | . 2 ⊢ (Fun 𝐹 → (dom 𝐹 ∈ 𝐵 → 𝐹 ∈ V)) |
3 | rnexg 7595 | . 2 ⊢ (𝐹 ∈ V → ran 𝐹 ∈ V) | |
4 | 2, 3 | syl6com 37 | 1 ⊢ (dom 𝐹 ∈ 𝐵 → (Fun 𝐹 → ran 𝐹 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 Vcvv 3441 dom cdm 5519 ran crn 5520 Fun wfun 6318 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 |
This theorem is referenced by: zfrep6 7638 fornex 7639 tz7.48-3 8063 inf0 9068 axcc2lem 9847 zorn2lem4 9910 fnct 9948 |
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