Theorem dnnumch3 43665

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 6085	. . . . 5 ⊢ (◡𝐹 “ {𝑥}) ⊆ dom 𝐹
2		dnnumch.f	. . . . . . 7 ⊢ 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))
3	2	tfr1 8383	. . . . . 6 ⊢ 𝐹 Fn On
4	3	fndmi 6640	. . . . 5 ⊢ dom 𝐹 = On
5	1, 4	sseqtri 3993	. . . 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 43663	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ran 𝐹)
9	8	sselda 3945	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ran 𝐹)
10		inisegn0 6101	. . . . 5 ⊢ (𝑥 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝑥}) ≠ ∅)
11	9, 10	sylib 221	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (◡𝐹 “ {𝑥}) ≠ ∅)
12		oninton 7793	. . . 4 ⊢ (((◡𝐹 “ {𝑥}) ⊆ On ∧ (◡𝐹 “ {𝑥}) ≠ ∅) → ∩ (◡𝐹 “ {𝑥}) ∈ On)
13	5, 11, 12	sylancr 598	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∩ (◡𝐹 “ {𝑥}) ∈ On)
14	13	fmpttd 7111	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})):𝐴⟶On)
15	2, 6, 7	dnnumch3lem 43664	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) = ∩ (◡𝐹 “ {𝑣}))
16	15	adantrr 729	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) = ∩ (◡𝐹 “ {𝑣}))
17	2, 6, 7	dnnumch3lem 43664	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) = ∩ (◡𝐹 “ {𝑤}))
18	17	adantrl 728	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) = ∩ (◡𝐹 “ {𝑤}))
19	16, 18	eqeq12d 2785	. . . 4 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) = ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) ↔ ∩ (◡𝐹 “ {𝑣}) = ∩ (◡𝐹 “ {𝑤})))
20		fveq2 6882	. . . . . . 7 ⊢ (∩ (◡𝐹 “ {𝑣}) = ∩ (◡𝐹 “ {𝑤}) → (𝐹‘∩ (◡𝐹 “ {𝑣})) = (𝐹‘∩ (◡𝐹 “ {𝑤})))
21	20	adantl 486	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ ∩ (◡𝐹 “ {𝑣}) = ∩ (◡𝐹 “ {𝑤})) → (𝐹‘∩ (◡𝐹 “ {𝑣})) = (𝐹‘∩ (◡𝐹 “ {𝑤})))
22		cnvimass 6085	. . . . . . . . . . 11 ⊢ (◡𝐹 “ {𝑣}) ⊆ dom 𝐹
23	22, 4	sseqtri 3993	. . . . . . . . . 10 ⊢ (◡𝐹 “ {𝑣}) ⊆ On
24	8	sselda 3945	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → 𝑣 ∈ ran 𝐹)
25		inisegn0 6101	. . . . . . . . . . 11 ⊢ (𝑣 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝑣}) ≠ ∅)
26	24, 25	sylib 221	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (◡𝐹 “ {𝑣}) ≠ ∅)
27		onint 7788	. . . . . . . . . 10 ⊢ (((◡𝐹 “ {𝑣}) ⊆ On ∧ (◡𝐹 “ {𝑣}) ≠ ∅) → ∩ (◡𝐹 “ {𝑣}) ∈ (◡𝐹 “ {𝑣}))
28	23, 26, 27	sylancr 598	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → ∩ (◡𝐹 “ {𝑣}) ∈ (◡𝐹 “ {𝑣}))
29		fniniseg 7056	. . . . . . . . . . 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 502	. . . . . . . . 9 ⊢ (∩ (◡𝐹 “ {𝑣}) ∈ (◡𝐹 “ {𝑣}) → (𝐹‘∩ (◡𝐹 “ {𝑣})) = 𝑣)
32	28, 31	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (𝐹‘∩ (◡𝐹 “ {𝑣})) = 𝑣)
33	32	adantrr 729	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝐹‘∩ (◡𝐹 “ {𝑣})) = 𝑣)
34	33	adantr 485	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ ∩ (◡𝐹 “ {𝑣}) = ∩ (◡𝐹 “ {𝑤})) → (𝐹‘∩ (◡𝐹 “ {𝑣})) = 𝑣)
35		cnvimass 6085	. . . . . . . . . . 11 ⊢ (◡𝐹 “ {𝑤}) ⊆ dom 𝐹
36	35, 4	sseqtri 3993	. . . . . . . . . 10 ⊢ (◡𝐹 “ {𝑤}) ⊆ On
37	8	sselda 3945	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ ran 𝐹)
38		inisegn0 6101	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝑤}) ≠ ∅)
39	37, 38	sylib 221	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (◡𝐹 “ {𝑤}) ≠ ∅)
40		onint 7788	. . . . . . . . . 10 ⊢ (((◡𝐹 “ {𝑤}) ⊆ On ∧ (◡𝐹 “ {𝑤}) ≠ ∅) → ∩ (◡𝐹 “ {𝑤}) ∈ (◡𝐹 “ {𝑤}))
41	36, 39, 40	sylancr 598	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ∩ (◡𝐹 “ {𝑤}) ∈ (◡𝐹 “ {𝑤}))
42		fniniseg 7056	. . . . . . . . . . 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 502	. . . . . . . . 9 ⊢ (∩ (◡𝐹 “ {𝑤}) ∈ (◡𝐹 “ {𝑤}) → (𝐹‘∩ (◡𝐹 “ {𝑤})) = 𝑤)
45	41, 44	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘∩ (◡𝐹 “ {𝑤})) = 𝑤)
46	45	adantrl 728	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝐹‘∩ (◡𝐹 “ {𝑤})) = 𝑤)
47	46	adantr 485	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ ∩ (◡𝐹 “ {𝑣}) = ∩ (◡𝐹 “ {𝑤})) → (𝐹‘∩ (◡𝐹 “ {𝑤})) = 𝑤)
48	21, 34, 47	3eqtr3d 2812	. . . . 5 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ ∩ (◡𝐹 “ {𝑣}) = ∩ (◡𝐹 “ {𝑤})) → 𝑣 = 𝑤)
49	48	ex 417	. . . 4 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (∩ (◡𝐹 “ {𝑣}) = ∩ (◡𝐹 “ {𝑤}) → 𝑣 = 𝑤))
50	19, 49	sylbid 243	. . 3 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) = ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) → 𝑣 = 𝑤))
51	50	ralrimivva 3214	. 2 ⊢ (𝜑 → ∀𝑣 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) = ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) → 𝑣 = 𝑤))
52		dff13 7253	. 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 594	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})):𝐴–1-1→On)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  Vcvv 3463   ∖ cdif 3910   ⊆ wss 3913  ∅c0 4294  𝒫 cpw 4567  {csn 4594  ∩ cint 4916   ↦ cmpt 5196  ^◡ccnv 5661  dom cdm 5662  ran crn 5663   “ cima 5665  Oncon0 6361   Fn wfn 6532  ⟶wf 6533  –1-1→wf1 6534  ‘cfv 6537  recscrecs 8356

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357

This theorem is referenced by:  dnwech  43666