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Theorem wunfunc 17849
Description: A weak universe is closed under the functor set operation. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof shortened by AV, 13-Oct-2024.)
Hypotheses
Ref Expression
wunfunc.1 (πœ‘ β†’ π‘ˆ ∈ WUni)
wunfunc.2 (πœ‘ β†’ 𝐢 ∈ π‘ˆ)
wunfunc.3 (πœ‘ β†’ 𝐷 ∈ π‘ˆ)
Assertion
Ref Expression
wunfunc (πœ‘ β†’ (𝐢 Func 𝐷) ∈ π‘ˆ)

Proof of Theorem wunfunc
Dummy variables 𝑓 𝑔 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wunfunc.1 . 2 (πœ‘ β†’ π‘ˆ ∈ WUni)
2 baseid 17147 . . . . 5 Base = Slot (Baseβ€˜ndx)
3 wunfunc.3 . . . . 5 (πœ‘ β†’ 𝐷 ∈ π‘ˆ)
42, 1, 3wunstr 17121 . . . 4 (πœ‘ β†’ (Baseβ€˜π·) ∈ π‘ˆ)
5 wunfunc.2 . . . . 5 (πœ‘ β†’ 𝐢 ∈ π‘ˆ)
62, 1, 5wunstr 17121 . . . 4 (πœ‘ β†’ (Baseβ€˜πΆ) ∈ π‘ˆ)
71, 4, 6wunmap 10721 . . 3 (πœ‘ β†’ ((Baseβ€˜π·) ↑m (Baseβ€˜πΆ)) ∈ π‘ˆ)
8 homid 17357 . . . . . . . . 9 Hom = Slot (Hom β€˜ndx)
98, 1, 5wunstr 17121 . . . . . . . 8 (πœ‘ β†’ (Hom β€˜πΆ) ∈ π‘ˆ)
101, 9wunrn 10724 . . . . . . 7 (πœ‘ β†’ ran (Hom β€˜πΆ) ∈ π‘ˆ)
111, 10wununi 10701 . . . . . 6 (πœ‘ β†’ βˆͺ ran (Hom β€˜πΆ) ∈ π‘ˆ)
128, 1, 3wunstr 17121 . . . . . . . 8 (πœ‘ β†’ (Hom β€˜π·) ∈ π‘ˆ)
131, 12wunrn 10724 . . . . . . 7 (πœ‘ β†’ ran (Hom β€˜π·) ∈ π‘ˆ)
141, 13wununi 10701 . . . . . 6 (πœ‘ β†’ βˆͺ ran (Hom β€˜π·) ∈ π‘ˆ)
151, 11, 14wunxp 10719 . . . . 5 (πœ‘ β†’ (βˆͺ ran (Hom β€˜πΆ) Γ— βˆͺ ran (Hom β€˜π·)) ∈ π‘ˆ)
161, 15wunpw 10702 . . . 4 (πœ‘ β†’ 𝒫 (βˆͺ ran (Hom β€˜πΆ) Γ— βˆͺ ran (Hom β€˜π·)) ∈ π‘ˆ)
171, 6, 6wunxp 10719 . . . 4 (πœ‘ β†’ ((Baseβ€˜πΆ) Γ— (Baseβ€˜πΆ)) ∈ π‘ˆ)
181, 16, 17wunmap 10721 . . 3 (πœ‘ β†’ (𝒫 (βˆͺ ran (Hom β€˜πΆ) Γ— βˆͺ ran (Hom β€˜π·)) ↑m ((Baseβ€˜πΆ) Γ— (Baseβ€˜πΆ))) ∈ π‘ˆ)
191, 7, 18wunxp 10719 . 2 (πœ‘ β†’ (((Baseβ€˜π·) ↑m (Baseβ€˜πΆ)) Γ— (𝒫 (βˆͺ ran (Hom β€˜πΆ) Γ— βˆͺ ran (Hom β€˜π·)) ↑m ((Baseβ€˜πΆ) Γ— (Baseβ€˜πΆ)))) ∈ π‘ˆ)
20 relfunc 17812 . . . 4 Rel (𝐢 Func 𝐷)
2120a1i 11 . . 3 (πœ‘ β†’ Rel (𝐢 Func 𝐷))
22 df-br 5150 . . . 4 (𝑓(𝐢 Func 𝐷)𝑔 ↔ βŸ¨π‘“, π‘”βŸ© ∈ (𝐢 Func 𝐷))
23 eqid 2733 . . . . . . . 8 (Baseβ€˜πΆ) = (Baseβ€˜πΆ)
24 eqid 2733 . . . . . . . 8 (Baseβ€˜π·) = (Baseβ€˜π·)
25 simpr 486 . . . . . . . 8 ((πœ‘ ∧ 𝑓(𝐢 Func 𝐷)𝑔) β†’ 𝑓(𝐢 Func 𝐷)𝑔)
2623, 24, 25funcf1 17816 . . . . . . 7 ((πœ‘ ∧ 𝑓(𝐢 Func 𝐷)𝑔) β†’ 𝑓:(Baseβ€˜πΆ)⟢(Baseβ€˜π·))
27 fvex 6905 . . . . . . . 8 (Baseβ€˜π·) ∈ V
28 fvex 6905 . . . . . . . 8 (Baseβ€˜πΆ) ∈ V
2927, 28elmap 8865 . . . . . . 7 (𝑓 ∈ ((Baseβ€˜π·) ↑m (Baseβ€˜πΆ)) ↔ 𝑓:(Baseβ€˜πΆ)⟢(Baseβ€˜π·))
3026, 29sylibr 233 . . . . . 6 ((πœ‘ ∧ 𝑓(𝐢 Func 𝐷)𝑔) β†’ 𝑓 ∈ ((Baseβ€˜π·) ↑m (Baseβ€˜πΆ)))
31 mapsspw 8872 . . . . . . . . . . 11 (((π‘“β€˜(1st β€˜π‘§))(Hom β€˜π·)(π‘“β€˜(2nd β€˜π‘§))) ↑m ((Hom β€˜πΆ)β€˜π‘§)) βŠ† 𝒫 (((Hom β€˜πΆ)β€˜π‘§) Γ— ((π‘“β€˜(1st β€˜π‘§))(Hom β€˜π·)(π‘“β€˜(2nd β€˜π‘§))))
32 fvssunirn 6925 . . . . . . . . . . . . 13 ((Hom β€˜πΆ)β€˜π‘§) βŠ† βˆͺ ran (Hom β€˜πΆ)
33 ovssunirn 7445 . . . . . . . . . . . . 13 ((π‘“β€˜(1st β€˜π‘§))(Hom β€˜π·)(π‘“β€˜(2nd β€˜π‘§))) βŠ† βˆͺ ran (Hom β€˜π·)
34 xpss12 5692 . . . . . . . . . . . . 13 ((((Hom β€˜πΆ)β€˜π‘§) βŠ† βˆͺ ran (Hom β€˜πΆ) ∧ ((π‘“β€˜(1st β€˜π‘§))(Hom β€˜π·)(π‘“β€˜(2nd β€˜π‘§))) βŠ† βˆͺ ran (Hom β€˜π·)) β†’ (((Hom β€˜πΆ)β€˜π‘§) Γ— ((π‘“β€˜(1st β€˜π‘§))(Hom β€˜π·)(π‘“β€˜(2nd β€˜π‘§)))) βŠ† (βˆͺ ran (Hom β€˜πΆ) Γ— βˆͺ ran (Hom β€˜π·)))
3532, 33, 34mp2an 691 . . . . . . . . . . . 12 (((Hom β€˜πΆ)β€˜π‘§) Γ— ((π‘“β€˜(1st β€˜π‘§))(Hom β€˜π·)(π‘“β€˜(2nd β€˜π‘§)))) βŠ† (βˆͺ ran (Hom β€˜πΆ) Γ— βˆͺ ran (Hom β€˜π·))
3635sspwi 4615 . . . . . . . . . . 11 𝒫 (((Hom β€˜πΆ)β€˜π‘§) Γ— ((π‘“β€˜(1st β€˜π‘§))(Hom β€˜π·)(π‘“β€˜(2nd β€˜π‘§)))) βŠ† 𝒫 (βˆͺ ran (Hom β€˜πΆ) Γ— βˆͺ ran (Hom β€˜π·))
3731, 36sstri 3992 . . . . . . . . . 10 (((π‘“β€˜(1st β€˜π‘§))(Hom β€˜π·)(π‘“β€˜(2nd β€˜π‘§))) ↑m ((Hom β€˜πΆ)β€˜π‘§)) βŠ† 𝒫 (βˆͺ ran (Hom β€˜πΆ) Γ— βˆͺ ran (Hom β€˜π·))
3837rgenw 3066 . . . . . . . . 9 βˆ€π‘§ ∈ ((Baseβ€˜πΆ) Γ— (Baseβ€˜πΆ))(((π‘“β€˜(1st β€˜π‘§))(Hom β€˜π·)(π‘“β€˜(2nd β€˜π‘§))) ↑m ((Hom β€˜πΆ)β€˜π‘§)) βŠ† 𝒫 (βˆͺ ran (Hom β€˜πΆ) Γ— βˆͺ ran (Hom β€˜π·))
39 ss2ixp 8904 . . . . . . . . 9 (βˆ€π‘§ ∈ ((Baseβ€˜πΆ) Γ— (Baseβ€˜πΆ))(((π‘“β€˜(1st β€˜π‘§))(Hom β€˜π·)(π‘“β€˜(2nd β€˜π‘§))) ↑m ((Hom β€˜πΆ)β€˜π‘§)) βŠ† 𝒫 (βˆͺ ran (Hom β€˜πΆ) Γ— βˆͺ ran (Hom β€˜π·)) β†’ X𝑧 ∈ ((Baseβ€˜πΆ) Γ— (Baseβ€˜πΆ))(((π‘“β€˜(1st β€˜π‘§))(Hom β€˜π·)(π‘“β€˜(2nd β€˜π‘§))) ↑m ((Hom β€˜πΆ)β€˜π‘§)) βŠ† X𝑧 ∈ ((Baseβ€˜πΆ) Γ— (Baseβ€˜πΆ))𝒫 (βˆͺ ran (Hom β€˜πΆ) Γ— βˆͺ ran (Hom β€˜π·)))
4038, 39ax-mp 5 . . . . . . . 8 X𝑧 ∈ ((Baseβ€˜πΆ) Γ— (Baseβ€˜πΆ))(((π‘“β€˜(1st β€˜π‘§))(Hom β€˜π·)(π‘“β€˜(2nd β€˜π‘§))) ↑m ((Hom β€˜πΆ)β€˜π‘§)) βŠ† X𝑧 ∈ ((Baseβ€˜πΆ) Γ— (Baseβ€˜πΆ))𝒫 (βˆͺ ran (Hom β€˜πΆ) Γ— βˆͺ ran (Hom β€˜π·))
4128, 28xpex 7740 . . . . . . . . 9 ((Baseβ€˜πΆ) Γ— (Baseβ€˜πΆ)) ∈ V
42 fvex 6905 . . . . . . . . . . . . 13 (Hom β€˜πΆ) ∈ V
4342rnex 7903 . . . . . . . . . . . 12 ran (Hom β€˜πΆ) ∈ V
4443uniex 7731 . . . . . . . . . . 11 βˆͺ ran (Hom β€˜πΆ) ∈ V
45 fvex 6905 . . . . . . . . . . . . 13 (Hom β€˜π·) ∈ V
4645rnex 7903 . . . . . . . . . . . 12 ran (Hom β€˜π·) ∈ V
4746uniex 7731 . . . . . . . . . . 11 βˆͺ ran (Hom β€˜π·) ∈ V
4844, 47xpex 7740 . . . . . . . . . 10 (βˆͺ ran (Hom β€˜πΆ) Γ— βˆͺ ran (Hom β€˜π·)) ∈ V
4948pwex 5379 . . . . . . . . 9 𝒫 (βˆͺ ran (Hom β€˜πΆ) Γ— βˆͺ ran (Hom β€˜π·)) ∈ V
5041, 49ixpconst 8901 . . . . . . . 8 X𝑧 ∈ ((Baseβ€˜πΆ) Γ— (Baseβ€˜πΆ))𝒫 (βˆͺ ran (Hom β€˜πΆ) Γ— βˆͺ ran (Hom β€˜π·)) = (𝒫 (βˆͺ ran (Hom β€˜πΆ) Γ— βˆͺ ran (Hom β€˜π·)) ↑m ((Baseβ€˜πΆ) Γ— (Baseβ€˜πΆ)))
5140, 50sseqtri 4019 . . . . . . 7 X𝑧 ∈ ((Baseβ€˜πΆ) Γ— (Baseβ€˜πΆ))(((π‘“β€˜(1st β€˜π‘§))(Hom β€˜π·)(π‘“β€˜(2nd β€˜π‘§))) ↑m ((Hom β€˜πΆ)β€˜π‘§)) βŠ† (𝒫 (βˆͺ ran (Hom β€˜πΆ) Γ— βˆͺ ran (Hom β€˜π·)) ↑m ((Baseβ€˜πΆ) Γ— (Baseβ€˜πΆ)))
52 eqid 2733 . . . . . . . 8 (Hom β€˜πΆ) = (Hom β€˜πΆ)
53 eqid 2733 . . . . . . . 8 (Hom β€˜π·) = (Hom β€˜π·)
5423, 52, 53, 25funcixp 17817 . . . . . . 7 ((πœ‘ ∧ 𝑓(𝐢 Func 𝐷)𝑔) β†’ 𝑔 ∈ X𝑧 ∈ ((Baseβ€˜πΆ) Γ— (Baseβ€˜πΆ))(((π‘“β€˜(1st β€˜π‘§))(Hom β€˜π·)(π‘“β€˜(2nd β€˜π‘§))) ↑m ((Hom β€˜πΆ)β€˜π‘§)))
5551, 54sselid 3981 . . . . . 6 ((πœ‘ ∧ 𝑓(𝐢 Func 𝐷)𝑔) β†’ 𝑔 ∈ (𝒫 (βˆͺ ran (Hom β€˜πΆ) Γ— βˆͺ ran (Hom β€˜π·)) ↑m ((Baseβ€˜πΆ) Γ— (Baseβ€˜πΆ))))
5630, 55opelxpd 5716 . . . . 5 ((πœ‘ ∧ 𝑓(𝐢 Func 𝐷)𝑔) β†’ βŸ¨π‘“, π‘”βŸ© ∈ (((Baseβ€˜π·) ↑m (Baseβ€˜πΆ)) Γ— (𝒫 (βˆͺ ran (Hom β€˜πΆ) Γ— βˆͺ ran (Hom β€˜π·)) ↑m ((Baseβ€˜πΆ) Γ— (Baseβ€˜πΆ)))))
5756ex 414 . . . 4 (πœ‘ β†’ (𝑓(𝐢 Func 𝐷)𝑔 β†’ βŸ¨π‘“, π‘”βŸ© ∈ (((Baseβ€˜π·) ↑m (Baseβ€˜πΆ)) Γ— (𝒫 (βˆͺ ran (Hom β€˜πΆ) Γ— βˆͺ ran (Hom β€˜π·)) ↑m ((Baseβ€˜πΆ) Γ— (Baseβ€˜πΆ))))))
5822, 57biimtrrid 242 . . 3 (πœ‘ β†’ (βŸ¨π‘“, π‘”βŸ© ∈ (𝐢 Func 𝐷) β†’ βŸ¨π‘“, π‘”βŸ© ∈ (((Baseβ€˜π·) ↑m (Baseβ€˜πΆ)) Γ— (𝒫 (βˆͺ ran (Hom β€˜πΆ) Γ— βˆͺ ran (Hom β€˜π·)) ↑m ((Baseβ€˜πΆ) Γ— (Baseβ€˜πΆ))))))
5921, 58relssdv 5789 . 2 (πœ‘ β†’ (𝐢 Func 𝐷) βŠ† (((Baseβ€˜π·) ↑m (Baseβ€˜πΆ)) Γ— (𝒫 (βˆͺ ran (Hom β€˜πΆ) Γ— βˆͺ ran (Hom β€˜π·)) ↑m ((Baseβ€˜πΆ) Γ— (Baseβ€˜πΆ)))))
601, 19, 59wunss 10707 1 (πœ‘ β†’ (𝐢 Func 𝐷) ∈ π‘ˆ)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∈ wcel 2107  βˆ€wral 3062   βŠ† wss 3949  π’« cpw 4603  βŸ¨cop 4635  βˆͺ cuni 4909   class class class wbr 5149   Γ— cxp 5675  ran crn 5678  Rel wrel 5682  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  1st c1st 7973  2nd c2nd 7974   ↑m cmap 8820  Xcixp 8891  WUnicwun 10695  ndxcnx 17126  Basecbs 17144  Hom chom 17208   Func cfunc 17804
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-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-map 8822  df-pm 8823  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-wun 10697  df-pnf 11250  df-mnf 11251  df-ltxr 11253  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-7 12280  df-8 12281  df-9 12282  df-n0 12473  df-dec 12678  df-slot 17115  df-ndx 17127  df-base 17145  df-hom 17221  df-func 17808
This theorem is referenced by:  wunnat  17907  wunnatOLD  17908  catcfuccl  18069  catcfucclOLD  18070
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