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| 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 7246 is proven without the Axiom of Replacement ax-rep 5279, but depends on ax-un 7755, which is not required for the proof of fex 7246. (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 7770 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × 𝐵) ∈ V) | |
| 2 | 1 | 3adant1 1131 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × 𝐵) ∈ V) |
| 3 | fssxp 6763 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (𝐴 × 𝐵)) | |
| 4 | 3 | 3ad2ant1 1134 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐹 ⊆ (𝐴 × 𝐵)) |
| 5 | 2, 4 | ssexd 5324 | 1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐹 ∈ V) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1087 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 × cxp 5683 ⟶wf 6557 |
| 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-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| 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-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-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-fun 6563 df-fn 6564 df-f 6565 |
| This theorem is referenced by: elmapg 8879 f1oen2g 9009 f1dom2g 9010 dom3d 9034 domssex2 9177 domssex 9178 mapxpen 9183 oismo 9580 wdomima2g 9626 dfac8clem 10072 acni2 10086 acnlem 10088 dfac4 10162 dfac2a 10170 axdc2lem 10488 axdc4lem 10495 axcclem 10497 mpoaddex 13030 addex 13031 mpomulex 13032 mulex 13033 seqf1olem2 14083 seqf1o 14084 limsuple 15514 limsuplt 15515 limsupbnd1 15518 caucvgrlem 15709 prdsplusg 17503 prdsmulr 17504 prdsvsca 17505 prdshom 17512 gsumval 18690 frmdplusg 18867 isghm 19233 odinf 19581 staffval 20842 cnfldcj 21373 cnfldds 21376 cnfldcjOLD 21386 cnflddsOLD 21389 xrsadd 21397 xrsmul 21398 xrsds 21427 ocvfval 21684 cnpfval 23242 iscnp2 23247 fmf 23953 tsmsval 24139 blfvalps 24393 nmfval 24601 tngnm 24672 tngngp2 24673 tngngpd 24674 tngngp 24675 nmoffn 24732 nmofval 24735 ishtpy 25004 tcphex 25251 elno 27690 adjeu 31908 ismeas 34200 isismty 37808 rrnval 37834 subex 42288 absex 42289 cjex 42290 sn-isghm 42683 |
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