Theorem dnnumch3 40788

Description: Define an injection from a set into the ordinals using 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
dnnumch3	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})):𝐴–1-1→On)

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 dnnumch3

Dummy variables 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnvimass 5978	. . . . 5 ⊢ (◡𝐹 “ {𝑥}) ⊆ dom 𝐹
2		dnnumch.f	. . . . . . 7 ⊢ 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))
3	2	tfr1 8199	. . . . . 6 ⊢ 𝐹 Fn On
4	3	fndmi 6521	. . . . 5 ⊢ dom 𝐹 = On
5	1, 4	sseqtri 3953	. . . 4 ⊢ (◡𝐹 “ {𝑥}) ⊆ On
6		dnnumch.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
7		dnnumch.g	. . . . . . 7 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺‘𝑦) ∈ 𝑦))
8	2, 6, 7	dnnumch2 40786	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ran 𝐹)
9	8	sselda 3917	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ran 𝐹)
10		inisegn0 5995	. . . . 5 ⊢ (𝑥 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝑥}) ≠ ∅)
11	9, 10	sylib 217	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (◡𝐹 “ {𝑥}) ≠ ∅)
12		oninton 7622	. . . 4 ⊢ (((◡𝐹 “ {𝑥}) ⊆ On ∧ (◡𝐹 “ {𝑥}) ≠ ∅) → ∩ (◡𝐹 “ {𝑥}) ∈ On)
13	5, 11, 12	sylancr 586	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∩ (◡𝐹 “ {𝑥}) ∈ On)
14	13	fmpttd 6971	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})):𝐴⟶On)
15	2, 6, 7	dnnumch3lem 40787	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) = ∩ (◡𝐹 “ {𝑣}))
16	15	adantrr 713	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) = ∩ (◡𝐹 “ {𝑣}))
17	2, 6, 7	dnnumch3lem 40787	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) = ∩ (◡𝐹 “ {𝑤}))
18	17	adantrl 712	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) = ∩ (◡𝐹 “ {𝑤}))
19	16, 18	eqeq12d 2754	. . . 4 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) = ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) ↔ ∩ (◡𝐹 “ {𝑣}) = ∩ (◡𝐹 “ {𝑤})))
20		fveq2 6756	. . . . . . 7 ⊢ (∩ (◡𝐹 “ {𝑣}) = ∩ (◡𝐹 “ {𝑤}) → (𝐹‘∩ (◡𝐹 “ {𝑣})) = (𝐹‘∩ (◡𝐹 “ {𝑤})))
21	20	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ ∩ (◡𝐹 “ {𝑣}) = ∩ (◡𝐹 “ {𝑤})) → (𝐹‘∩ (◡𝐹 “ {𝑣})) = (𝐹‘∩ (◡𝐹 “ {𝑤})))
22		cnvimass 5978	. . . . . . . . . . 11 ⊢ (◡𝐹 “ {𝑣}) ⊆ dom 𝐹
23	22, 4	sseqtri 3953	. . . . . . . . . 10 ⊢ (◡𝐹 “ {𝑣}) ⊆ On
24	8	sselda 3917	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → 𝑣 ∈ ran 𝐹)
25		inisegn0 5995	. . . . . . . . . . 11 ⊢ (𝑣 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝑣}) ≠ ∅)
26	24, 25	sylib 217	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (◡𝐹 “ {𝑣}) ≠ ∅)
27		onint 7617	. . . . . . . . . 10 ⊢ (((◡𝐹 “ {𝑣}) ⊆ On ∧ (◡𝐹 “ {𝑣}) ≠ ∅) → ∩ (◡𝐹 “ {𝑣}) ∈ (◡𝐹 “ {𝑣}))
28	23, 26, 27	sylancr 586	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → ∩ (◡𝐹 “ {𝑣}) ∈ (◡𝐹 “ {𝑣}))
29		fniniseg 6919	. . . . . . . . . . 11 ⊢ (𝐹 Fn On → (∩ (◡𝐹 “ {𝑣}) ∈ (◡𝐹 “ {𝑣}) ↔ (∩ (◡𝐹 “ {𝑣}) ∈ On ∧ (𝐹‘∩ (◡𝐹 “ {𝑣})) = 𝑣)))
30	3, 29	ax-mp 5	. . . . . . . . . 10 ⊢ (∩ (◡𝐹 “ {𝑣}) ∈ (◡𝐹 “ {𝑣}) ↔ (∩ (◡𝐹 “ {𝑣}) ∈ On ∧ (𝐹‘∩ (◡𝐹 “ {𝑣})) = 𝑣))
31	30	simprbi 496	. . . . . . . . 9 ⊢ (∩ (◡𝐹 “ {𝑣}) ∈ (◡𝐹 “ {𝑣}) → (𝐹‘∩ (◡𝐹 “ {𝑣})) = 𝑣)
32	28, 31	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (𝐹‘∩ (◡𝐹 “ {𝑣})) = 𝑣)
33	32	adantrr 713	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝐹‘∩ (◡𝐹 “ {𝑣})) = 𝑣)
34	33	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ ∩ (◡𝐹 “ {𝑣}) = ∩ (◡𝐹 “ {𝑤})) → (𝐹‘∩ (◡𝐹 “ {𝑣})) = 𝑣)
35		cnvimass 5978	. . . . . . . . . . 11 ⊢ (◡𝐹 “ {𝑤}) ⊆ dom 𝐹
36	35, 4	sseqtri 3953	. . . . . . . . . 10 ⊢ (◡𝐹 “ {𝑤}) ⊆ On
37	8	sselda 3917	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ ran 𝐹)
38		inisegn0 5995	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝑤}) ≠ ∅)
39	37, 38	sylib 217	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (◡𝐹 “ {𝑤}) ≠ ∅)
40		onint 7617	. . . . . . . . . 10 ⊢ (((◡𝐹 “ {𝑤}) ⊆ On ∧ (◡𝐹 “ {𝑤}) ≠ ∅) → ∩ (◡𝐹 “ {𝑤}) ∈ (◡𝐹 “ {𝑤}))
41	36, 39, 40	sylancr 586	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ∩ (◡𝐹 “ {𝑤}) ∈ (◡𝐹 “ {𝑤}))
42		fniniseg 6919	. . . . . . . . . . 11 ⊢ (𝐹 Fn On → (∩ (◡𝐹 “ {𝑤}) ∈ (◡𝐹 “ {𝑤}) ↔ (∩ (◡𝐹 “ {𝑤}) ∈ On ∧ (𝐹‘∩ (◡𝐹 “ {𝑤})) = 𝑤)))
43	3, 42	ax-mp 5	. . . . . . . . . 10 ⊢ (∩ (◡𝐹 “ {𝑤}) ∈ (◡𝐹 “ {𝑤}) ↔ (∩ (◡𝐹 “ {𝑤}) ∈ On ∧ (𝐹‘∩ (◡𝐹 “ {𝑤})) = 𝑤))
44	43	simprbi 496	. . . . . . . . 9 ⊢ (∩ (◡𝐹 “ {𝑤}) ∈ (◡𝐹 “ {𝑤}) → (𝐹‘∩ (◡𝐹 “ {𝑤})) = 𝑤)
45	41, 44	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘∩ (◡𝐹 “ {𝑤})) = 𝑤)
46	45	adantrl 712	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝐹‘∩ (◡𝐹 “ {𝑤})) = 𝑤)
47	46	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ ∩ (◡𝐹 “ {𝑣}) = ∩ (◡𝐹 “ {𝑤})) → (𝐹‘∩ (◡𝐹 “ {𝑤})) = 𝑤)
48	21, 34, 47	3eqtr3d 2786	. . . . 5 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ ∩ (◡𝐹 “ {𝑣}) = ∩ (◡𝐹 “ {𝑤})) → 𝑣 = 𝑤)
49	48	ex 412	. . . 4 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (∩ (◡𝐹 “ {𝑣}) = ∩ (◡𝐹 “ {𝑤}) → 𝑣 = 𝑤))
50	19, 49	sylbid 239	. . 3 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) = ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) → 𝑣 = 𝑤))
51	50	ralrimivva 3114	. 2 ⊢ (𝜑 → ∀𝑣 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) = ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) → 𝑣 = 𝑤))
52		dff13 7109	. 2 ⊢ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})):𝐴–1-1→On ↔ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})):𝐴⟶On ∧ ∀𝑣 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) = ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) → 𝑣 = 𝑤)))
53	14, 51, 52	sylanbrc 582	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})):𝐴–1-1→On)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  Vcvv 3422   ∖ cdif 3880   ⊆ wss 3883  ∅c0 4253  𝒫 cpw 4530  {csn 4558  ∩ cint 4876   ↦ cmpt 5153  ^◡ccnv 5579  dom cdm 5580  ran crn 5581   “ cima 5583  Oncon0 6251   Fn wfn 6413  ⟶wf 6414  –1-1→wf1 6415  ‘cfv 6418  recscrecs 8172

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173

This theorem is referenced by:  dnwech  40789