![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 5290. 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 6645 | . 2 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ V) → (𝐴 “ 𝐵) ∈ V) | |
3 | 1, 2 | mpan2 689 | 1 ⊢ (Fun 𝐴 → (𝐴 “ 𝐵) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 Vcvv 3462 “ cima 5685 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-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pr 5433 |
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-sb 2061 df-mo 2529 df-clab 2704 df-cleq 2718 df-clel 2803 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 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-br 5154 df-opab 5216 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-fun 6556 |
This theorem is referenced by: isarep2 6650 isofr 7354 isose 7355 f1opw 7682 f1oweALT 7986 ttrclse 9770 tz9.12lem2 9831 hsmexlem4 10472 hsmexlem5 10473 zorn2lem7 10545 uniimadom 10587 zexALT 12630 psdmul 22160 fbasrn 23879 oldf 27881 negsproplem2 28038 precsexlem10 28215 seqsex 28259 noseqex 28263 zsex 28330 dimval 33495 dimvalfi 33496 fnwe2lem2 42712 setrec2fun 48438 |
Copyright terms: Public domain | W3C validator |