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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 4994. 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 6208 | . 2 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ V) → (𝐴 “ 𝐵) ∈ V) | |
3 | 1, 2 | mpan2 684 | 1 ⊢ (Fun 𝐴 → (𝐴 “ 𝐵) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2166 Vcvv 3414 “ cima 5345 Fun wfun 6117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pr 5127 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-br 4874 df-opab 4936 df-id 5250 df-xp 5348 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-fun 6125 |
This theorem is referenced by: isarep2 6211 isofr 6847 isose 6848 f1opw 7149 f1oweALT 7412 tz9.12lem2 8928 hsmexlem4 9566 hsmexlem5 9567 zorn2lem7 9639 uniimadom 9681 zexALT 11723 fbasrn 22058 fnwe2lem2 38464 setrec2fun 43334 |
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