![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fex2 | Structured version Visualization version GIF version |
Description: A function with bounded domain and codomain is a set. This version of fex 7207 is proven without the Axiom of Replacement ax-rep 5273, but depends on ax-un 7703, which is not required for the proof of fex 7207. (Contributed by Mario Carneiro, 24-Jun-2015.) |
Ref | Expression |
---|---|
fex2 | ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐹 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpexg 7715 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × 𝐵) ∈ V) | |
2 | 1 | 3adant1 1130 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × 𝐵) ∈ V) |
3 | fssxp 6727 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (𝐴 × 𝐵)) | |
4 | 3 | 3ad2ant1 1133 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐹 ⊆ (𝐴 × 𝐵)) |
5 | 2, 4 | ssexd 5312 | 1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐹 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1087 ∈ wcel 2106 Vcvv 3469 ⊆ wss 3939 × cxp 5662 ⟶wf 6523 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 ax-sep 5287 ax-nul 5294 ax-pow 5351 ax-pr 5415 ax-un 7703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3429 df-v 3471 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4314 df-if 4518 df-pw 4593 df-sn 4618 df-pr 4620 df-op 4624 df-uni 4897 df-br 5137 df-opab 5199 df-xp 5670 df-rel 5671 df-cnv 5672 df-dm 5674 df-rn 5675 df-fun 6529 df-fn 6530 df-f 6531 |
This theorem is referenced by: elmapg 8811 f1oen2g 8942 f1dom2g 8943 f1dom2gOLD 8944 dom3d 8968 domssex2 9115 domssex 9116 mapxpen 9121 oismo 9512 wdomima2g 9558 dfac8clem 10004 acni2 10018 acnlem 10020 dfac4 10094 dfac2a 10101 axdc2lem 10420 axdc4lem 10427 axcclem 10429 addex 12949 mulex 12950 seqf1olem2 13985 seqf1o 13986 limsuple 15399 limsuplt 15400 limsupbnd1 15403 caucvgrlem 15596 prdsplusg 17381 prdsmulr 17382 prdsvsca 17383 prdshom 17390 gsumval 18573 frmdplusg 18705 odinf 19390 staffval 20397 cnfldcj 20878 cnfldds 20881 xrsadd 20889 xrsmul 20890 xrsds 20915 ocvfval 21145 cnpfval 22660 iscnp2 22665 fmf 23371 tsmsval 23557 blfvalps 23811 nmfval 24019 tngnm 24090 tngngp2 24091 tngngpd 24092 tngngp 24093 nmoffn 24150 nmofval 24153 ishtpy 24410 tcphex 24656 adjeu 30951 ismeas 32963 isismty 36409 rrnval 36435 |
Copyright terms: Public domain | W3C validator |