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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 7236. (Contributed by NM, 11-Nov-1995.) |
Ref | Expression |
---|---|
funrnex | ⊢ (dom 𝐹 ∈ 𝐵 → (Fun 𝐹 → ran 𝐹 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funex 7236 | . . 3 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ 𝐵) → 𝐹 ∈ V) | |
2 | 1 | ex 411 | . 2 ⊢ (Fun 𝐹 → (dom 𝐹 ∈ 𝐵 → 𝐹 ∈ V)) |
3 | rnexg 7915 | . 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 2099 Vcvv 3462 dom cdm 5682 ran crn 5683 Fun wfun 6548 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 |
This theorem is referenced by: zfrep6 7968 focdmex 7969 tz7.48-3 8474 inf0 9664 axcc2lem 10479 zorn2lem4 10542 fnct 10580 tfsconcatrev 43014 |
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