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Theorem dnnumch2 38117
Description: Define an enumeration (weak dominance version) of a set from a choice function. (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
dnnumch2 (𝜑𝐴 ⊆ ran 𝐹)
Distinct variable groups:   𝑦,𝐹   𝑦,𝐺,𝑧   𝑦,𝐴,𝑧
Allowed substitution hints:   𝜑(𝑦,𝑧)   𝐹(𝑧)   𝑉(𝑦,𝑧)

Proof of Theorem dnnumch2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dnnumch.f . . 3 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))
2 dnnumch.a . . 3 (𝜑𝐴𝑉)
3 dnnumch.g . . 3 (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺𝑦) ∈ 𝑦))
41, 2, 3dnnumch1 38116 . 2 (𝜑 → ∃𝑥 ∈ On (𝐹𝑥):𝑥1-1-onto𝐴)
5 f1ofo 6305 . . . . . 6 ((𝐹𝑥):𝑥1-1-onto𝐴 → (𝐹𝑥):𝑥onto𝐴)
6 forn 6279 . . . . . 6 ((𝐹𝑥):𝑥onto𝐴 → ran (𝐹𝑥) = 𝐴)
75, 6syl 17 . . . . 5 ((𝐹𝑥):𝑥1-1-onto𝐴 → ran (𝐹𝑥) = 𝐴)
8 resss 5580 . . . . . 6 (𝐹𝑥) ⊆ 𝐹
9 rnss 5509 . . . . . 6 ((𝐹𝑥) ⊆ 𝐹 → ran (𝐹𝑥) ⊆ ran 𝐹)
108, 9mp1i 13 . . . . 5 ((𝐹𝑥):𝑥1-1-onto𝐴 → ran (𝐹𝑥) ⊆ ran 𝐹)
117, 10eqsstr3d 3781 . . . 4 ((𝐹𝑥):𝑥1-1-onto𝐴𝐴 ⊆ ran 𝐹)
1211a1i 11 . . 3 (𝜑 → ((𝐹𝑥):𝑥1-1-onto𝐴𝐴 ⊆ ran 𝐹))
1312rexlimdvw 3172 . 2 (𝜑 → (∃𝑥 ∈ On (𝐹𝑥):𝑥1-1-onto𝐴𝐴 ⊆ ran 𝐹))
144, 13mpd 15 1 (𝜑𝐴 ⊆ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  wne 2932  wral 3050  wrex 3051  Vcvv 3340  cdif 3712  wss 3715  c0 4058  𝒫 cpw 4302  cmpt 4881  ran crn 5267  cres 5268  Oncon0 5884  ontowfo 6047  1-1-ontowf1o 6048  cfv 6049  recscrecs 7636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-wrecs 7576  df-recs 7637
This theorem is referenced by:  dnnumch3lem  38118  dnnumch3  38119
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