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Theorem dnnumch1 41120
Description: Define an enumeration of a set from a choice function; second part, it restricts to a bijection. EDITORIAL: overlaps dfac8a 9879. (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 8297 . . . . . . 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 6820 . . . . . . 7 (𝐹𝑥) = (recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))‘𝑥)
53tfr1 8290 . . . . . . . . . . 11 𝐹 Fn On
6 fnfun 6579 . . . . . . . . . . 11 (𝐹 Fn On → Fun 𝐹)
75, 6ax-mp 5 . . . . . . . . . 10 Fun 𝐹
8 vex 3445 . . . . . . . . . 10 𝑥 ∈ V
9 resfunexg 7141 . . . . . . . . . 10 ((Fun 𝐹𝑥 ∈ V) → (𝐹𝑥) ∈ V)
107, 8, 9mp2an 689 . . . . . . . . 9 (𝐹𝑥) ∈ V
11 rneq 5871 . . . . . . . . . . . . 13 (𝑤 = (𝐹𝑥) → ran 𝑤 = ran (𝐹𝑥))
12 df-ima 5627 . . . . . . . . . . . . 13 (𝐹𝑥) = ran (𝐹𝑥)
1311, 12eqtr4di 2794 . . . . . . . . . . . 12 (𝑤 = (𝐹𝑥) → ran 𝑤 = (𝐹𝑥))
1413difeq2d 4068 . . . . . . . . . . 11 (𝑤 = (𝐹𝑥) → (𝐴 ∖ ran 𝑤) = (𝐴 ∖ (𝐹𝑥)))
1514fveq2d 6823 . . . . . . . . . 10 (𝑤 = (𝐹𝑥) → (𝐺‘(𝐴 ∖ ran 𝑤)) = (𝐺‘(𝐴 ∖ (𝐹𝑥))))
16 rneq 5871 . . . . . . . . . . . . 13 (𝑧 = 𝑤 → ran 𝑧 = ran 𝑤)
1716difeq2d 4068 . . . . . . . . . . . 12 (𝑧 = 𝑤 → (𝐴 ∖ ran 𝑧) = (𝐴 ∖ ran 𝑤))
1817fveq2d 6823 . . . . . . . . . . 11 (𝑧 = 𝑤 → (𝐺‘(𝐴 ∖ ran 𝑧)) = (𝐺‘(𝐴 ∖ ran 𝑤)))
1918cbvmptv 5202 . . . . . . . . . 10 (𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))) = (𝑤 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑤)))
20 fvex 6832 . . . . . . . . . 10 (𝐺‘(𝐴 ∖ (𝐹𝑥))) ∈ V
2115, 19, 20fvmpt 6925 . . . . . . . . 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 5913 . . . . . . . . 9 (𝐹𝑥) = (recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))) ↾ 𝑥)
2423fveq2i 6822 . . . . . . . 8 ((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))‘(𝐹𝑥)) = ((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))‘(recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))) ↾ 𝑥))
2522, 24eqtr3i 2766 . . . . . . 7 (𝐺‘(𝐴 ∖ (𝐹𝑥))) = ((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))‘(recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))) ↾ 𝑥))
262, 4, 253eqtr4g 2801 . . . . . 6 (𝑥 ∈ On → (𝐹𝑥) = (𝐺‘(𝐴 ∖ (𝐹𝑥))))
2726ad2antlr 724 . . . . 5 (((𝜑𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹𝑥)) ≠ ∅) → (𝐹𝑥) = (𝐺‘(𝐴 ∖ (𝐹𝑥))))
28 difss 4077 . . . . . . . . 9 (𝐴 ∖ (𝐹𝑥)) ⊆ 𝐴
29 elpw2g 5285 . . . . . . . . . 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 3003 . . . . . . . . . 10 (𝑦 = (𝐴 ∖ (𝐹𝑥)) → (𝑦 ≠ ∅ ↔ (𝐴 ∖ (𝐹𝑥)) ≠ ∅))
34 fveq2 6819 . . . . . . . . . . 11 (𝑦 = (𝐴 ∖ (𝐹𝑥)) → (𝐺𝑦) = (𝐺‘(𝐴 ∖ (𝐹𝑥))))
35 id 22 . . . . . . . . . . 11 (𝑦 = (𝐴 ∖ (𝐹𝑥)) → 𝑦 = (𝐴 ∖ (𝐹𝑥)))
3634, 35eleq12d 2831 . . . . . . . . . 10 (𝑦 = (𝐴 ∖ (𝐹𝑥)) → ((𝐺𝑦) ∈ 𝑦 ↔ (𝐺‘(𝐴 ∖ (𝐹𝑥))) ∈ (𝐴 ∖ (𝐹𝑥))))
3733, 36imbi12d 344 . . . . . . . . 9 (𝑦 = (𝐴 ∖ (𝐹𝑥)) → ((𝑦 ≠ ∅ → (𝐺𝑦) ∈ 𝑦) ↔ ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐺‘(𝐴 ∖ (𝐹𝑥))) ∈ (𝐴 ∖ (𝐹𝑥)))))
3837rspcva 3568 . . . . . . . 8 (((𝐴 ∖ (𝐹𝑥)) ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺𝑦) ∈ 𝑦)) → ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐺‘(𝐴 ∖ (𝐹𝑥))) ∈ (𝐴 ∖ (𝐹𝑥))))
3931, 32, 38syl2anc 584 . . . . . . 7 (𝜑 → ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐺‘(𝐴 ∖ (𝐹𝑥))) ∈ (𝐴 ∖ (𝐹𝑥))))
4039adantr 481 . . . . . 6 ((𝜑𝑥 ∈ On) → ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐺‘(𝐴 ∖ (𝐹𝑥))) ∈ (𝐴 ∖ (𝐹𝑥))))
4140imp 407 . . . . 5 (((𝜑𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹𝑥)) ≠ ∅) → (𝐺‘(𝐴 ∖ (𝐹𝑥))) ∈ (𝐴 ∖ (𝐹𝑥)))
4227, 41eqeltrd 2837 . . . 4 (((𝜑𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹𝑥)) ≠ ∅) → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)))
4342ex 413 . . 3 ((𝜑𝑥 ∈ On) → ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))))
4443ralrimiva 3139 . 2 (𝜑 → ∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))))
455tz7.49c 8339 . 2 ((𝐴𝑉 ∧ ∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)))) → ∃𝑥 ∈ On (𝐹𝑥):𝑥1-1-onto𝐴)
461, 44, 45syl2anc 584 1 (𝜑 → ∃𝑥 ∈ On (𝐹𝑥):𝑥1-1-onto𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1540  wcel 2105  wne 2940  wral 3061  wrex 3070  Vcvv 3441  cdif 3894  wss 3897  c0 4268  𝒫 cpw 4546  cmpt 5172  ran crn 5615  cres 5616  cima 5617  Oncon0 6296  Fun wfun 6467   Fn wfn 6468  1-1-ontowf1o 6472  cfv 6473  recscrecs 8263
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5226  ax-sep 5240  ax-nul 5247  ax-pr 5369  ax-un 7642
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-int 4894  df-iun 4940  df-br 5090  df-opab 5152  df-mpt 5173  df-tr 5207  df-id 5512  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5569  df-we 5571  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6232  df-ord 6299  df-on 6300  df-suc 6302  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-ov 7332  df-2nd 7892  df-frecs 8159  df-wrecs 8190  df-recs 8264
This theorem is referenced by:  dnnumch2  41121
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