Theorem dnnumch2 37615

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	⊢ (𝜑 → 𝐴 ∈ 𝑉)
dnnumch.g	⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺‘𝑦) ∈ 𝑦))

Assertion

Ref	Expression
dnnumch2	⊢ (𝜑 → 𝐴 ⊆ 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.

Step	Hyp	Ref	Expression
1		dnnumch.f	. . 3 ⊢ 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))
2		dnnumch.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		dnnumch.g	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺‘𝑦) ∈ 𝑦))
4	1, 2, 3	dnnumch1 37614	. 2 ⊢ (𝜑 → ∃𝑥 ∈ On (𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴)
5		f1ofo 6144	. . . . . 6 ⊢ ((𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴 → (𝐹 ↾ 𝑥):𝑥–onto→𝐴)
6		forn 6118	. . . . . 6 ⊢ ((𝐹 ↾ 𝑥):𝑥–onto→𝐴 → ran (𝐹 ↾ 𝑥) = 𝐴)
7	5, 6	syl 17	. . . . 5 ⊢ ((𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴 → ran (𝐹 ↾ 𝑥) = 𝐴)
8		resss 5422	. . . . . 6 ⊢ (𝐹 ↾ 𝑥) ⊆ 𝐹
9		rnss 5354	. . . . . 6 ⊢ ((𝐹 ↾ 𝑥) ⊆ 𝐹 → ran (𝐹 ↾ 𝑥) ⊆ ran 𝐹)
10	8, 9	mp1i 13	. . . . 5 ⊢ ((𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴 → ran (𝐹 ↾ 𝑥) ⊆ ran 𝐹)
11	7, 10	eqsstr3d 3640	. . . 4 ⊢ ((𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴 → 𝐴 ⊆ ran 𝐹)
12	11	a1i 11	. . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴 → 𝐴 ⊆ ran 𝐹))
13	12	rexlimdvw 3034	. 2 ⊢ (𝜑 → (∃𝑥 ∈ On (𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴 → 𝐴 ⊆ ran 𝐹))
14	4, 13	mpd 15	1 ⊢ (𝜑 → 𝐴 ⊆ ran 𝐹)

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158   ↦ cmpt 4729  ran crn 5115   ↾ cres 5116  Oncon0 5723  –onto→wfo 5886  –1-1-onto→wf1o 5887  ‘cfv 5888  recscrecs 7467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-wrecs 7407  df-recs 7468

This theorem is referenced by:  dnnumch3lem  37616  dnnumch3  37617