Theorem dnnumch3 43043

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 6056	. . . . 5 ⊢ (◡𝐹 “ {𝑥}) ⊆ dom 𝐹
2		dnnumch.f	. . . . . . 7 ⊢ 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))
3	2	tfr1 8368	. . . . . 6 ⊢ 𝐹 Fn On
4	3	fndmi 6625	. . . . 5 ⊢ dom 𝐹 = On
5	1, 4	sseqtri 3998	. . . 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 43041	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ran 𝐹)
9	8	sselda 3949	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ran 𝐹)
10		inisegn0 6072	. . . . 5 ⊢ (𝑥 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝑥}) ≠ ∅)
11	9, 10	sylib 218	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (◡𝐹 “ {𝑥}) ≠ ∅)
12		oninton 7774	. . . 4 ⊢ (((◡𝐹 “ {𝑥}) ⊆ On ∧ (◡𝐹 “ {𝑥}) ≠ ∅) → ∩ (◡𝐹 “ {𝑥}) ∈ On)
13	5, 11, 12	sylancr 587	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∩ (◡𝐹 “ {𝑥}) ∈ On)
14	13	fmpttd 7090	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})):𝐴⟶On)
15	2, 6, 7	dnnumch3lem 43042	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) = ∩ (◡𝐹 “ {𝑣}))
16	15	adantrr 717	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) = ∩ (◡𝐹 “ {𝑣}))
17	2, 6, 7	dnnumch3lem 43042	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) = ∩ (◡𝐹 “ {𝑤}))
18	17	adantrl 716	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) = ∩ (◡𝐹 “ {𝑤}))
19	16, 18	eqeq12d 2746	. . . 4 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) = ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) ↔ ∩ (◡𝐹 “ {𝑣}) = ∩ (◡𝐹 “ {𝑤})))
20		fveq2 6861	. . . . . . 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 6056	. . . . . . . . . . 11 ⊢ (◡𝐹 “ {𝑣}) ⊆ dom 𝐹
23	22, 4	sseqtri 3998	. . . . . . . . . 10 ⊢ (◡𝐹 “ {𝑣}) ⊆ On
24	8	sselda 3949	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → 𝑣 ∈ ran 𝐹)
25		inisegn0 6072	. . . . . . . . . . 11 ⊢ (𝑣 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝑣}) ≠ ∅)
26	24, 25	sylib 218	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (◡𝐹 “ {𝑣}) ≠ ∅)
27		onint 7769	. . . . . . . . . 10 ⊢ (((◡𝐹 “ {𝑣}) ⊆ On ∧ (◡𝐹 “ {𝑣}) ≠ ∅) → ∩ (◡𝐹 “ {𝑣}) ∈ (◡𝐹 “ {𝑣}))
28	23, 26, 27	sylancr 587	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → ∩ (◡𝐹 “ {𝑣}) ∈ (◡𝐹 “ {𝑣}))
29		fniniseg 7035	. . . . . . . . . . 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 717	. . . . . . 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 6056	. . . . . . . . . . 11 ⊢ (◡𝐹 “ {𝑤}) ⊆ dom 𝐹
36	35, 4	sseqtri 3998	. . . . . . . . . 10 ⊢ (◡𝐹 “ {𝑤}) ⊆ On
37	8	sselda 3949	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ ran 𝐹)
38		inisegn0 6072	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝑤}) ≠ ∅)
39	37, 38	sylib 218	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (◡𝐹 “ {𝑤}) ≠ ∅)
40		onint 7769	. . . . . . . . . 10 ⊢ (((◡𝐹 “ {𝑤}) ⊆ On ∧ (◡𝐹 “ {𝑤}) ≠ ∅) → ∩ (◡𝐹 “ {𝑤}) ∈ (◡𝐹 “ {𝑤}))
41	36, 39, 40	sylancr 587	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ∩ (◡𝐹 “ {𝑤}) ∈ (◡𝐹 “ {𝑤}))
42		fniniseg 7035	. . . . . . . . . . 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 716	. . . . . . 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 2773	. . . . 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 240	. . 3 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) = ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) → 𝑣 = 𝑤))
51	50	ralrimivva 3181	. 2 ⊢ (𝜑 → ∀𝑣 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) = ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) → 𝑣 = 𝑤))
52		dff13 7232	. 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 583	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})):𝐴–1-1→On)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  ∀wral 3045  Vcvv 3450   ∖ cdif 3914   ⊆ wss 3917  ∅c0 4299  𝒫 cpw 4566  {csn 4592  ∩ cint 4913   ↦ cmpt 5191  ^◡ccnv 5640  dom cdm 5641  ran crn 5642   “ cima 5644  Oncon0 6335   Fn wfn 6509  ⟶wf 6510  –1-1→wf1 6511  ‘cfv 6514  recscrecs 8342

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343

This theorem is referenced by:  dnwech  43044