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| Mirrors > Home > MPE Home > Th. List > funimaex | Structured version Visualization version GIF version | ||
| Description: The image of a set under any function is also a set. Equivalent of Axiom of Replacement ax-rep 5279. Axiom 39(vi) of [Quine] p. 284. Compare Exercise 9 of [TakeutiZaring] p. 29. (Contributed by NM, 17-Nov-2002.) |
| Ref | Expression |
|---|---|
| zfrep5.1 | ⊢ 𝐵 ∈ V |
| Ref | Expression |
|---|---|
| funimaex | ⊢ (Fun 𝐴 → (𝐴 “ 𝐵) ∈ V) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | zfrep5.1 | . 2 ⊢ 𝐵 ∈ V | |
| 2 | funimaexg 6653 | . 2 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ V) → (𝐴 “ 𝐵) ∈ V) | |
| 3 | 1, 2 | mpan2 691 | 1 ⊢ (Fun 𝐴 → (𝐴 “ 𝐵) ∈ V) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 Vcvv 3480 “ cima 5688 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-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| 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-mo 2540 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-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-fun 6563 |
| This theorem is referenced by: isarep2 6658 isofr 7362 isose 7363 f1opw 7689 f1oweALT 7997 ttrclse 9767 tz9.12lem2 9828 hsmexlem4 10469 hsmexlem5 10470 zorn2lem7 10542 uniimadom 10584 zexALT 12633 psdmul 22170 fbasrn 23892 oldf 27896 madefi 27950 negsproplem2 28061 precsexlem10 28240 seqsex 28291 noseqex 28295 zsex 28366 dimval 33651 dimvalfi 33652 fnwe2lem2 43063 relpfr 44975 setrec2fun 49211 |
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