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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 7177 is proven without the Axiom of Replacement ax-rep 5243, but depends on ax-un 7673, which is not required for the proof of fex 7177. (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 7685 | . . 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 6697 | . . 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 5282 | 1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐹 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1088 ∈ wcel 2107 Vcvv 3444 ⊆ wss 3911 × cxp 5632 ⟶wf 6493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-xp 5640 df-rel 5641 df-cnv 5642 df-dm 5644 df-rn 5645 df-fun 6499 df-fn 6500 df-f 6501 |
This theorem is referenced by: elmapg 8781 f1oen2g 8911 f1dom2g 8912 f1dom2gOLD 8913 dom3d 8937 domssex2 9084 domssex 9085 mapxpen 9090 oismo 9481 wdomima2g 9527 dfac8clem 9973 acni2 9987 acnlem 9989 dfac4 10063 dfac2a 10070 axdc2lem 10389 axdc4lem 10396 axcclem 10398 addex 12918 mulex 12919 seqf1olem2 13954 seqf1o 13955 limsuple 15366 limsuplt 15367 limsupbnd1 15370 caucvgrlem 15563 prdsplusg 17345 prdsmulr 17346 prdsvsca 17347 prdshom 17354 gsumval 18537 frmdplusg 18669 odinf 19350 staffval 20320 cnfldcj 20819 cnfldds 20822 xrsadd 20830 xrsmul 20831 xrsds 20856 ocvfval 21086 cnpfval 22601 iscnp2 22606 fmf 23312 tsmsval 23498 blfvalps 23752 nmfval 23960 tngnm 24031 tngngp2 24032 tngngpd 24033 tngngp 24034 nmoffn 24091 nmofval 24094 ishtpy 24351 tcphex 24597 adjeu 30873 ismeas 32855 isismty 36306 rrnval 36332 |
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