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 5182. 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 6434 | . 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 2110 Vcvv 3494 “ cima 5552 Fun wfun 6343 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-fun 6351 |
This theorem is referenced by: isarep2 6437 isofr 7089 isose 7090 f1opw 7395 f1oweALT 7667 tz9.12lem2 9211 hsmexlem4 9845 hsmexlem5 9846 zorn2lem7 9918 uniimadom 9960 zexALT 11995 fbasrn 22486 dimval 30996 dimvalfi 30997 fnwe2lem2 39644 setrec2fun 44789 |
Copyright terms: Public domain | W3C validator |