Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dnnumch1 Structured version   Visualization version   GIF version

Theorem dnnumch1 38985
Description: Define an enumeration of a set from a choice function; second part, it restricts to a bijection. EDITORIAL: overlaps dfac8a 9242. (Contributed by Stefan O'Rear, 18-Jan-2015.)
Hypotheses
Ref Expression
dnnumch.f 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))
dnnumch.a (𝜑𝐴𝑉)
dnnumch.g (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺𝑦) ∈ 𝑦))
Assertion
Ref Expression
dnnumch1 (𝜑 → ∃𝑥 ∈ On (𝐹𝑥):𝑥1-1-onto𝐴)
Distinct variable groups:   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦,𝑧   𝑥,𝐴,𝑦,𝑧   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑦,𝑧)   𝐹(𝑧)   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem dnnumch1
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 dnnumch.a . 2 (𝜑𝐴𝑉)
2 recsval 7837 . . . . . . 7 (𝑥 ∈ On → (recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))‘𝑥) = ((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))‘(recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))) ↾ 𝑥)))
3 dnnumch.f . . . . . . . 8 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))
43fveq1i 6494 . . . . . . 7 (𝐹𝑥) = (recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))‘𝑥)
53tfr1 7830 . . . . . . . . . . 11 𝐹 Fn On
6 fnfun 6280 . . . . . . . . . . 11 (𝐹 Fn On → Fun 𝐹)
75, 6ax-mp 5 . . . . . . . . . 10 Fun 𝐹
8 vex 3412 . . . . . . . . . 10 𝑥 ∈ V
9 resfunexg 6798 . . . . . . . . . 10 ((Fun 𝐹𝑥 ∈ V) → (𝐹𝑥) ∈ V)
107, 8, 9mp2an 679 . . . . . . . . 9 (𝐹𝑥) ∈ V
11 rneq 5642 . . . . . . . . . . . . 13 (𝑤 = (𝐹𝑥) → ran 𝑤 = ran (𝐹𝑥))
12 df-ima 5413 . . . . . . . . . . . . 13 (𝐹𝑥) = ran (𝐹𝑥)
1311, 12syl6eqr 2826 . . . . . . . . . . . 12 (𝑤 = (𝐹𝑥) → ran 𝑤 = (𝐹𝑥))
1413difeq2d 3985 . . . . . . . . . . 11 (𝑤 = (𝐹𝑥) → (𝐴 ∖ ran 𝑤) = (𝐴 ∖ (𝐹𝑥)))
1514fveq2d 6497 . . . . . . . . . 10 (𝑤 = (𝐹𝑥) → (𝐺‘(𝐴 ∖ ran 𝑤)) = (𝐺‘(𝐴 ∖ (𝐹𝑥))))
16 rneq 5642 . . . . . . . . . . . . 13 (𝑧 = 𝑤 → ran 𝑧 = ran 𝑤)
1716difeq2d 3985 . . . . . . . . . . . 12 (𝑧 = 𝑤 → (𝐴 ∖ ran 𝑧) = (𝐴 ∖ ran 𝑤))
1817fveq2d 6497 . . . . . . . . . . 11 (𝑧 = 𝑤 → (𝐺‘(𝐴 ∖ ran 𝑧)) = (𝐺‘(𝐴 ∖ ran 𝑤)))
1918cbvmptv 5022 . . . . . . . . . 10 (𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))) = (𝑤 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑤)))
20 fvex 6506 . . . . . . . . . 10 (𝐺‘(𝐴 ∖ (𝐹𝑥))) ∈ V
2115, 19, 20fvmpt 6589 . . . . . . . . 9 ((𝐹𝑥) ∈ V → ((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))‘(𝐹𝑥)) = (𝐺‘(𝐴 ∖ (𝐹𝑥))))
2210, 21ax-mp 5 . . . . . . . 8 ((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))‘(𝐹𝑥)) = (𝐺‘(𝐴 ∖ (𝐹𝑥)))
233reseq1i 5684 . . . . . . . . 9 (𝐹𝑥) = (recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))) ↾ 𝑥)
2423fveq2i 6496 . . . . . . . 8 ((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))‘(𝐹𝑥)) = ((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))‘(recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))) ↾ 𝑥))
2522, 24eqtr3i 2798 . . . . . . 7 (𝐺‘(𝐴 ∖ (𝐹𝑥))) = ((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))‘(recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))) ↾ 𝑥))
262, 4, 253eqtr4g 2833 . . . . . 6 (𝑥 ∈ On → (𝐹𝑥) = (𝐺‘(𝐴 ∖ (𝐹𝑥))))
2726ad2antlr 714 . . . . 5 (((𝜑𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹𝑥)) ≠ ∅) → (𝐹𝑥) = (𝐺‘(𝐴 ∖ (𝐹𝑥))))
28 difss 3994 . . . . . . . . 9 (𝐴 ∖ (𝐹𝑥)) ⊆ 𝐴
29 elpw2g 5097 . . . . . . . . . 10 (𝐴𝑉 → ((𝐴 ∖ (𝐹𝑥)) ∈ 𝒫 𝐴 ↔ (𝐴 ∖ (𝐹𝑥)) ⊆ 𝐴))
301, 29syl 17 . . . . . . . . 9 (𝜑 → ((𝐴 ∖ (𝐹𝑥)) ∈ 𝒫 𝐴 ↔ (𝐴 ∖ (𝐹𝑥)) ⊆ 𝐴))
3128, 30mpbiri 250 . . . . . . . 8 (𝜑 → (𝐴 ∖ (𝐹𝑥)) ∈ 𝒫 𝐴)
32 dnnumch.g . . . . . . . 8 (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺𝑦) ∈ 𝑦))
33 neeq1 3023 . . . . . . . . . 10 (𝑦 = (𝐴 ∖ (𝐹𝑥)) → (𝑦 ≠ ∅ ↔ (𝐴 ∖ (𝐹𝑥)) ≠ ∅))
34 fveq2 6493 . . . . . . . . . . 11 (𝑦 = (𝐴 ∖ (𝐹𝑥)) → (𝐺𝑦) = (𝐺‘(𝐴 ∖ (𝐹𝑥))))
35 id 22 . . . . . . . . . . 11 (𝑦 = (𝐴 ∖ (𝐹𝑥)) → 𝑦 = (𝐴 ∖ (𝐹𝑥)))
3634, 35eleq12d 2854 . . . . . . . . . 10 (𝑦 = (𝐴 ∖ (𝐹𝑥)) → ((𝐺𝑦) ∈ 𝑦 ↔ (𝐺‘(𝐴 ∖ (𝐹𝑥))) ∈ (𝐴 ∖ (𝐹𝑥))))
3733, 36imbi12d 337 . . . . . . . . 9 (𝑦 = (𝐴 ∖ (𝐹𝑥)) → ((𝑦 ≠ ∅ → (𝐺𝑦) ∈ 𝑦) ↔ ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐺‘(𝐴 ∖ (𝐹𝑥))) ∈ (𝐴 ∖ (𝐹𝑥)))))
3837rspcva 3527 . . . . . . . 8 (((𝐴 ∖ (𝐹𝑥)) ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺𝑦) ∈ 𝑦)) → ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐺‘(𝐴 ∖ (𝐹𝑥))) ∈ (𝐴 ∖ (𝐹𝑥))))
3931, 32, 38syl2anc 576 . . . . . . 7 (𝜑 → ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐺‘(𝐴 ∖ (𝐹𝑥))) ∈ (𝐴 ∖ (𝐹𝑥))))
4039adantr 473 . . . . . 6 ((𝜑𝑥 ∈ On) → ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐺‘(𝐴 ∖ (𝐹𝑥))) ∈ (𝐴 ∖ (𝐹𝑥))))
4140imp 398 . . . . 5 (((𝜑𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹𝑥)) ≠ ∅) → (𝐺‘(𝐴 ∖ (𝐹𝑥))) ∈ (𝐴 ∖ (𝐹𝑥)))
4227, 41eqeltrd 2860 . . . 4 (((𝜑𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹𝑥)) ≠ ∅) → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)))
4342ex 405 . . 3 ((𝜑𝑥 ∈ On) → ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))))
4443ralrimiva 3126 . 2 (𝜑 → ∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))))
455tz7.49c 7878 . 2 ((𝐴𝑉 ∧ ∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)))) → ∃𝑥 ∈ On (𝐹𝑥):𝑥1-1-onto𝐴)
461, 44, 45syl2anc 576 1 (𝜑 → ∃𝑥 ∈ On (𝐹𝑥):𝑥1-1-onto𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387   = wceq 1507  wcel 2048  wne 2961  wral 3082  wrex 3083  Vcvv 3409  cdif 3822  wss 3825  c0 4173  𝒫 cpw 4416  cmpt 5002  ran crn 5401  cres 5402  cima 5403  Oncon0 6023  Fun wfun 6176   Fn wfn 6177  1-1-ontowf1o 6181  cfv 6182  recscrecs 7804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-pss 3841  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-int 4744  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5305  df-eprel 5310  df-po 5319  df-so 5320  df-fr 5359  df-we 5361  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-pred 5980  df-ord 6026  df-on 6027  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-wrecs 7743  df-recs 7805
This theorem is referenced by:  dnnumch2  38986
  Copyright terms: Public domain W3C validator