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Theorem dnnumch1 42742
Description: Define an enumeration of a set from a choice function; second part, it restricts to a bijection. EDITORIAL: overlaps dfac8a 10066. (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 8426 . . . . . . 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 6894 . . . . . . 7 (𝐹𝑥) = (recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))‘𝑥)
53tfr1 8419 . . . . . . . . . . 11 𝐹 Fn On
6 fnfun 6652 . . . . . . . . . . 11 (𝐹 Fn On → Fun 𝐹)
75, 6ax-mp 5 . . . . . . . . . 10 Fun 𝐹
8 vex 3466 . . . . . . . . . 10 𝑥 ∈ V
9 resfunexg 7224 . . . . . . . . . 10 ((Fun 𝐹𝑥 ∈ V) → (𝐹𝑥) ∈ V)
107, 8, 9mp2an 690 . . . . . . . . 9 (𝐹𝑥) ∈ V
11 rneq 5934 . . . . . . . . . . . . 13 (𝑤 = (𝐹𝑥) → ran 𝑤 = ran (𝐹𝑥))
12 df-ima 5687 . . . . . . . . . . . . 13 (𝐹𝑥) = ran (𝐹𝑥)
1311, 12eqtr4di 2784 . . . . . . . . . . . 12 (𝑤 = (𝐹𝑥) → ran 𝑤 = (𝐹𝑥))
1413difeq2d 4118 . . . . . . . . . . 11 (𝑤 = (𝐹𝑥) → (𝐴 ∖ ran 𝑤) = (𝐴 ∖ (𝐹𝑥)))
1514fveq2d 6897 . . . . . . . . . 10 (𝑤 = (𝐹𝑥) → (𝐺‘(𝐴 ∖ ran 𝑤)) = (𝐺‘(𝐴 ∖ (𝐹𝑥))))
16 rneq 5934 . . . . . . . . . . . . 13 (𝑧 = 𝑤 → ran 𝑧 = ran 𝑤)
1716difeq2d 4118 . . . . . . . . . . . 12 (𝑧 = 𝑤 → (𝐴 ∖ ran 𝑧) = (𝐴 ∖ ran 𝑤))
1817fveq2d 6897 . . . . . . . . . . 11 (𝑧 = 𝑤 → (𝐺‘(𝐴 ∖ ran 𝑧)) = (𝐺‘(𝐴 ∖ ran 𝑤)))
1918cbvmptv 5258 . . . . . . . . . 10 (𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))) = (𝑤 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑤)))
20 fvex 6906 . . . . . . . . . 10 (𝐺‘(𝐴 ∖ (𝐹𝑥))) ∈ V
2115, 19, 20fvmpt 7001 . . . . . . . . 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 5977 . . . . . . . . 9 (𝐹𝑥) = (recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))) ↾ 𝑥)
2423fveq2i 6896 . . . . . . . 8 ((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))‘(𝐹𝑥)) = ((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))‘(recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))) ↾ 𝑥))
2522, 24eqtr3i 2756 . . . . . . 7 (𝐺‘(𝐴 ∖ (𝐹𝑥))) = ((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))‘(recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))) ↾ 𝑥))
262, 4, 253eqtr4g 2791 . . . . . 6 (𝑥 ∈ On → (𝐹𝑥) = (𝐺‘(𝐴 ∖ (𝐹𝑥))))
2726ad2antlr 725 . . . . 5 (((𝜑𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹𝑥)) ≠ ∅) → (𝐹𝑥) = (𝐺‘(𝐴 ∖ (𝐹𝑥))))
28 difss 4128 . . . . . . . . 9 (𝐴 ∖ (𝐹𝑥)) ⊆ 𝐴
29 elpw2g 5343 . . . . . . . . . 10 (𝐴𝑉 → ((𝐴 ∖ (𝐹𝑥)) ∈ 𝒫 𝐴 ↔ (𝐴 ∖ (𝐹𝑥)) ⊆ 𝐴))
301, 29syl 17 . . . . . . . . 9 (𝜑 → ((𝐴 ∖ (𝐹𝑥)) ∈ 𝒫 𝐴 ↔ (𝐴 ∖ (𝐹𝑥)) ⊆ 𝐴))
3128, 30mpbiri 257 . . . . . . . 8 (𝜑 → (𝐴 ∖ (𝐹𝑥)) ∈ 𝒫 𝐴)
32 dnnumch.g . . . . . . . 8 (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺𝑦) ∈ 𝑦))
33 neeq1 2993 . . . . . . . . . 10 (𝑦 = (𝐴 ∖ (𝐹𝑥)) → (𝑦 ≠ ∅ ↔ (𝐴 ∖ (𝐹𝑥)) ≠ ∅))
34 fveq2 6893 . . . . . . . . . . 11 (𝑦 = (𝐴 ∖ (𝐹𝑥)) → (𝐺𝑦) = (𝐺‘(𝐴 ∖ (𝐹𝑥))))
35 id 22 . . . . . . . . . . 11 (𝑦 = (𝐴 ∖ (𝐹𝑥)) → 𝑦 = (𝐴 ∖ (𝐹𝑥)))
3634, 35eleq12d 2820 . . . . . . . . . 10 (𝑦 = (𝐴 ∖ (𝐹𝑥)) → ((𝐺𝑦) ∈ 𝑦 ↔ (𝐺‘(𝐴 ∖ (𝐹𝑥))) ∈ (𝐴 ∖ (𝐹𝑥))))
3733, 36imbi12d 343 . . . . . . . . 9 (𝑦 = (𝐴 ∖ (𝐹𝑥)) → ((𝑦 ≠ ∅ → (𝐺𝑦) ∈ 𝑦) ↔ ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐺‘(𝐴 ∖ (𝐹𝑥))) ∈ (𝐴 ∖ (𝐹𝑥)))))
3837rspcva 3605 . . . . . . . 8 (((𝐴 ∖ (𝐹𝑥)) ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺𝑦) ∈ 𝑦)) → ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐺‘(𝐴 ∖ (𝐹𝑥))) ∈ (𝐴 ∖ (𝐹𝑥))))
3931, 32, 38syl2anc 582 . . . . . . 7 (𝜑 → ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐺‘(𝐴 ∖ (𝐹𝑥))) ∈ (𝐴 ∖ (𝐹𝑥))))
4039adantr 479 . . . . . 6 ((𝜑𝑥 ∈ On) → ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐺‘(𝐴 ∖ (𝐹𝑥))) ∈ (𝐴 ∖ (𝐹𝑥))))
4140imp 405 . . . . 5 (((𝜑𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹𝑥)) ≠ ∅) → (𝐺‘(𝐴 ∖ (𝐹𝑥))) ∈ (𝐴 ∖ (𝐹𝑥)))
4227, 41eqeltrd 2826 . . . 4 (((𝜑𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹𝑥)) ≠ ∅) → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)))
4342ex 411 . . 3 ((𝜑𝑥 ∈ On) → ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))))
4443ralrimiva 3136 . 2 (𝜑 → ∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))))
455tz7.49c 8468 . 2 ((𝐴𝑉 ∧ ∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)))) → ∃𝑥 ∈ On (𝐹𝑥):𝑥1-1-onto𝐴)
461, 44, 45syl2anc 582 1 (𝜑 → ∃𝑥 ∈ On (𝐹𝑥):𝑥1-1-onto𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wcel 2099  wne 2930  wral 3051  wrex 3060  Vcvv 3462  cdif 3943  wss 3946  c0 4322  𝒫 cpw 4597  cmpt 5228  ran crn 5675  cres 5676  cima 5677  Oncon0 6368  Fun wfun 6540   Fn wfn 6541  1-1-ontowf1o 6545  cfv 6546  recscrecs 8392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pr 5425  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-int 4947  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-tr 5263  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6304  df-ord 6371  df-on 6372  df-suc 6374  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-ov 7419  df-2nd 7996  df-frecs 8288  df-wrecs 8319  df-recs 8393
This theorem is referenced by:  dnnumch2  42743
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