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Theorem dnnumch1 37931
Description: Define an enumeration of a set from a choice function; second part, it restricts to a bijection. EDITORIAL: overlaps dfac8a 8891. (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 7545 . . . . . . 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 6230 . . . . . . 7 (𝐹𝑥) = (recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))‘𝑥)
53tfr1 7538 . . . . . . . . . . 11 𝐹 Fn On
6 fnfun 6026 . . . . . . . . . . 11 (𝐹 Fn On → Fun 𝐹)
75, 6ax-mp 5 . . . . . . . . . 10 Fun 𝐹
8 vex 3234 . . . . . . . . . 10 𝑥 ∈ V
9 resfunexg 6520 . . . . . . . . . 10 ((Fun 𝐹𝑥 ∈ V) → (𝐹𝑥) ∈ V)
107, 8, 9mp2an 708 . . . . . . . . 9 (𝐹𝑥) ∈ V
11 rneq 5383 . . . . . . . . . . . . 13 (𝑤 = (𝐹𝑥) → ran 𝑤 = ran (𝐹𝑥))
12 df-ima 5156 . . . . . . . . . . . . 13 (𝐹𝑥) = ran (𝐹𝑥)
1311, 12syl6eqr 2703 . . . . . . . . . . . 12 (𝑤 = (𝐹𝑥) → ran 𝑤 = (𝐹𝑥))
1413difeq2d 3761 . . . . . . . . . . 11 (𝑤 = (𝐹𝑥) → (𝐴 ∖ ran 𝑤) = (𝐴 ∖ (𝐹𝑥)))
1514fveq2d 6233 . . . . . . . . . 10 (𝑤 = (𝐹𝑥) → (𝐺‘(𝐴 ∖ ran 𝑤)) = (𝐺‘(𝐴 ∖ (𝐹𝑥))))
16 rneq 5383 . . . . . . . . . . . . 13 (𝑧 = 𝑤 → ran 𝑧 = ran 𝑤)
1716difeq2d 3761 . . . . . . . . . . . 12 (𝑧 = 𝑤 → (𝐴 ∖ ran 𝑧) = (𝐴 ∖ ran 𝑤))
1817fveq2d 6233 . . . . . . . . . . 11 (𝑧 = 𝑤 → (𝐺‘(𝐴 ∖ ran 𝑧)) = (𝐺‘(𝐴 ∖ ran 𝑤)))
1918cbvmptv 4783 . . . . . . . . . 10 (𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))) = (𝑤 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑤)))
20 fvex 6239 . . . . . . . . . 10 (𝐺‘(𝐴 ∖ (𝐹𝑥))) ∈ V
2115, 19, 20fvmpt 6321 . . . . . . . . 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 5424 . . . . . . . . 9 (𝐹𝑥) = (recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))) ↾ 𝑥)
2423fveq2i 6232 . . . . . . . 8 ((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))‘(𝐹𝑥)) = ((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))‘(recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))) ↾ 𝑥))
2522, 24eqtr3i 2675 . . . . . . 7 (𝐺‘(𝐴 ∖ (𝐹𝑥))) = ((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))‘(recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))) ↾ 𝑥))
262, 4, 253eqtr4g 2710 . . . . . 6 (𝑥 ∈ On → (𝐹𝑥) = (𝐺‘(𝐴 ∖ (𝐹𝑥))))
2726ad2antlr 763 . . . . 5 (((𝜑𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹𝑥)) ≠ ∅) → (𝐹𝑥) = (𝐺‘(𝐴 ∖ (𝐹𝑥))))
28 difss 3770 . . . . . . . . 9 (𝐴 ∖ (𝐹𝑥)) ⊆ 𝐴
29 elpw2g 4857 . . . . . . . . . 10 (𝐴𝑉 → ((𝐴 ∖ (𝐹𝑥)) ∈ 𝒫 𝐴 ↔ (𝐴 ∖ (𝐹𝑥)) ⊆ 𝐴))
301, 29syl 17 . . . . . . . . 9 (𝜑 → ((𝐴 ∖ (𝐹𝑥)) ∈ 𝒫 𝐴 ↔ (𝐴 ∖ (𝐹𝑥)) ⊆ 𝐴))
3128, 30mpbiri 248 . . . . . . . 8 (𝜑 → (𝐴 ∖ (𝐹𝑥)) ∈ 𝒫 𝐴)
32 dnnumch.g . . . . . . . 8 (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺𝑦) ∈ 𝑦))
33 neeq1 2885 . . . . . . . . . 10 (𝑦 = (𝐴 ∖ (𝐹𝑥)) → (𝑦 ≠ ∅ ↔ (𝐴 ∖ (𝐹𝑥)) ≠ ∅))
34 fveq2 6229 . . . . . . . . . . 11 (𝑦 = (𝐴 ∖ (𝐹𝑥)) → (𝐺𝑦) = (𝐺‘(𝐴 ∖ (𝐹𝑥))))
35 id 22 . . . . . . . . . . 11 (𝑦 = (𝐴 ∖ (𝐹𝑥)) → 𝑦 = (𝐴 ∖ (𝐹𝑥)))
3634, 35eleq12d 2724 . . . . . . . . . 10 (𝑦 = (𝐴 ∖ (𝐹𝑥)) → ((𝐺𝑦) ∈ 𝑦 ↔ (𝐺‘(𝐴 ∖ (𝐹𝑥))) ∈ (𝐴 ∖ (𝐹𝑥))))
3733, 36imbi12d 333 . . . . . . . . 9 (𝑦 = (𝐴 ∖ (𝐹𝑥)) → ((𝑦 ≠ ∅ → (𝐺𝑦) ∈ 𝑦) ↔ ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐺‘(𝐴 ∖ (𝐹𝑥))) ∈ (𝐴 ∖ (𝐹𝑥)))))
3837rspcva 3338 . . . . . . . 8 (((𝐴 ∖ (𝐹𝑥)) ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺𝑦) ∈ 𝑦)) → ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐺‘(𝐴 ∖ (𝐹𝑥))) ∈ (𝐴 ∖ (𝐹𝑥))))
3931, 32, 38syl2anc 694 . . . . . . 7 (𝜑 → ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐺‘(𝐴 ∖ (𝐹𝑥))) ∈ (𝐴 ∖ (𝐹𝑥))))
4039adantr 480 . . . . . 6 ((𝜑𝑥 ∈ On) → ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐺‘(𝐴 ∖ (𝐹𝑥))) ∈ (𝐴 ∖ (𝐹𝑥))))
4140imp 444 . . . . 5 (((𝜑𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹𝑥)) ≠ ∅) → (𝐺‘(𝐴 ∖ (𝐹𝑥))) ∈ (𝐴 ∖ (𝐹𝑥)))
4227, 41eqeltrd 2730 . . . 4 (((𝜑𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹𝑥)) ≠ ∅) → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)))
4342ex 449 . . 3 ((𝜑𝑥 ∈ On) → ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))))
4443ralrimiva 2995 . 2 (𝜑 → ∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))))
455tz7.49c 7586 . 2 ((𝐴𝑉 ∧ ∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)))) → ∃𝑥 ∈ On (𝐹𝑥):𝑥1-1-onto𝐴)
461, 44, 45syl2anc 694 1 (𝜑 → ∃𝑥 ∈ On (𝐹𝑥):𝑥1-1-onto𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  Vcvv 3231  cdif 3604  wss 3607  c0 3948  𝒫 cpw 4191  cmpt 4762  ran crn 5144  cres 5145  cima 5146  Oncon0 5761  Fun wfun 5920   Fn wfn 5921  1-1-ontowf1o 5925  cfv 5926  recscrecs 7512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-wrecs 7452  df-recs 7513
This theorem is referenced by:  dnnumch2  37932
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