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 38140
Description: Define an enumeration of a set from a choice function; second part, it restricts to a bijection. EDITORIAL: overlaps dfac8a 9053. (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 7653 . . . . . . 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 6333 . . . . . . 7 (𝐹𝑥) = (recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))‘𝑥)
53tfr1 7646 . . . . . . . . . . 11 𝐹 Fn On
6 fnfun 6128 . . . . . . . . . . 11 (𝐹 Fn On → Fun 𝐹)
75, 6ax-mp 5 . . . . . . . . . 10 Fun 𝐹
8 vex 3354 . . . . . . . . . 10 𝑥 ∈ V
9 resfunexg 6623 . . . . . . . . . 10 ((Fun 𝐹𝑥 ∈ V) → (𝐹𝑥) ∈ V)
107, 8, 9mp2an 664 . . . . . . . . 9 (𝐹𝑥) ∈ V
11 rneq 5489 . . . . . . . . . . . . 13 (𝑤 = (𝐹𝑥) → ran 𝑤 = ran (𝐹𝑥))
12 df-ima 5262 . . . . . . . . . . . . 13 (𝐹𝑥) = ran (𝐹𝑥)
1311, 12syl6eqr 2823 . . . . . . . . . . . 12 (𝑤 = (𝐹𝑥) → ran 𝑤 = (𝐹𝑥))
1413difeq2d 3879 . . . . . . . . . . 11 (𝑤 = (𝐹𝑥) → (𝐴 ∖ ran 𝑤) = (𝐴 ∖ (𝐹𝑥)))
1514fveq2d 6336 . . . . . . . . . 10 (𝑤 = (𝐹𝑥) → (𝐺‘(𝐴 ∖ ran 𝑤)) = (𝐺‘(𝐴 ∖ (𝐹𝑥))))
16 rneq 5489 . . . . . . . . . . . . 13 (𝑧 = 𝑤 → ran 𝑧 = ran 𝑤)
1716difeq2d 3879 . . . . . . . . . . . 12 (𝑧 = 𝑤 → (𝐴 ∖ ran 𝑧) = (𝐴 ∖ ran 𝑤))
1817fveq2d 6336 . . . . . . . . . . 11 (𝑧 = 𝑤 → (𝐺‘(𝐴 ∖ ran 𝑧)) = (𝐺‘(𝐴 ∖ ran 𝑤)))
1918cbvmptv 4884 . . . . . . . . . 10 (𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))) = (𝑤 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑤)))
20 fvex 6342 . . . . . . . . . 10 (𝐺‘(𝐴 ∖ (𝐹𝑥))) ∈ V
2115, 19, 20fvmpt 6424 . . . . . . . . 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 5530 . . . . . . . . 9 (𝐹𝑥) = (recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))) ↾ 𝑥)
2423fveq2i 6335 . . . . . . . 8 ((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))‘(𝐹𝑥)) = ((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))‘(recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))) ↾ 𝑥))
2522, 24eqtr3i 2795 . . . . . . 7 (𝐺‘(𝐴 ∖ (𝐹𝑥))) = ((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))‘(recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))) ↾ 𝑥))
262, 4, 253eqtr4g 2830 . . . . . 6 (𝑥 ∈ On → (𝐹𝑥) = (𝐺‘(𝐴 ∖ (𝐹𝑥))))
2726ad2antlr 698 . . . . 5 (((𝜑𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹𝑥)) ≠ ∅) → (𝐹𝑥) = (𝐺‘(𝐴 ∖ (𝐹𝑥))))
28 difss 3888 . . . . . . . . 9 (𝐴 ∖ (𝐹𝑥)) ⊆ 𝐴
29 elpw2g 4958 . . . . . . . . . 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 3005 . . . . . . . . . 10 (𝑦 = (𝐴 ∖ (𝐹𝑥)) → (𝑦 ≠ ∅ ↔ (𝐴 ∖ (𝐹𝑥)) ≠ ∅))
34 fveq2 6332 . . . . . . . . . . 11 (𝑦 = (𝐴 ∖ (𝐹𝑥)) → (𝐺𝑦) = (𝐺‘(𝐴 ∖ (𝐹𝑥))))
35 id 22 . . . . . . . . . . 11 (𝑦 = (𝐴 ∖ (𝐹𝑥)) → 𝑦 = (𝐴 ∖ (𝐹𝑥)))
3634, 35eleq12d 2844 . . . . . . . . . 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 3458 . . . . . . . 8 (((𝐴 ∖ (𝐹𝑥)) ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺𝑦) ∈ 𝑦)) → ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐺‘(𝐴 ∖ (𝐹𝑥))) ∈ (𝐴 ∖ (𝐹𝑥))))
3931, 32, 38syl2anc 565 . . . . . . 7 (𝜑 → ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐺‘(𝐴 ∖ (𝐹𝑥))) ∈ (𝐴 ∖ (𝐹𝑥))))
4039adantr 466 . . . . . 6 ((𝜑𝑥 ∈ On) → ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐺‘(𝐴 ∖ (𝐹𝑥))) ∈ (𝐴 ∖ (𝐹𝑥))))
4140imp 393 . . . . 5 (((𝜑𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹𝑥)) ≠ ∅) → (𝐺‘(𝐴 ∖ (𝐹𝑥))) ∈ (𝐴 ∖ (𝐹𝑥)))
4227, 41eqeltrd 2850 . . . 4 (((𝜑𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹𝑥)) ≠ ∅) → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)))
4342ex 397 . . 3 ((𝜑𝑥 ∈ On) → ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))))
4443ralrimiva 3115 . 2 (𝜑 → ∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))))
455tz7.49c 7694 . 2 ((𝐴𝑉 ∧ ∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)))) → ∃𝑥 ∈ On (𝐹𝑥):𝑥1-1-onto𝐴)
461, 44, 45syl2anc 565 1 (𝜑 → ∃𝑥 ∈ On (𝐹𝑥):𝑥1-1-onto𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wne 2943  wral 3061  wrex 3062  Vcvv 3351  cdif 3720  wss 3723  c0 4063  𝒫 cpw 4297  cmpt 4863  ran crn 5250  cres 5251  cima 5252  Oncon0 5866  Fun wfun 6025   Fn wfn 6026  1-1-ontowf1o 6030  cfv 6031  recscrecs 7620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-wrecs 7559  df-recs 7621
This theorem is referenced by:  dnnumch2  38141
  Copyright terms: Public domain W3C validator