Theorem dnnumch3 42745

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 6083	. . . . 5 ⊢ (◡𝐹 “ {𝑥}) ⊆ dom 𝐹
2		dnnumch.f	. . . . . . 7 ⊢ 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))
3	2	tfr1 8419	. . . . . 6 ⊢ 𝐹 Fn On
4	3	fndmi 6656	. . . . 5 ⊢ dom 𝐹 = On
5	1, 4	sseqtri 4015	. . . 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 42743	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ran 𝐹)
9	8	sselda 3978	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ran 𝐹)
10		inisegn0 6100	. . . . 5 ⊢ (𝑥 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝑥}) ≠ ∅)
11	9, 10	sylib 217	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (◡𝐹 “ {𝑥}) ≠ ∅)
12		oninton 7796	. . . 4 ⊢ (((◡𝐹 “ {𝑥}) ⊆ On ∧ (◡𝐹 “ {𝑥}) ≠ ∅) → ∩ (◡𝐹 “ {𝑥}) ∈ On)
13	5, 11, 12	sylancr 585	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∩ (◡𝐹 “ {𝑥}) ∈ On)
14	13	fmpttd 7121	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})):𝐴⟶On)
15	2, 6, 7	dnnumch3lem 42744	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) = ∩ (◡𝐹 “ {𝑣}))
16	15	adantrr 715	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) = ∩ (◡𝐹 “ {𝑣}))
17	2, 6, 7	dnnumch3lem 42744	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) = ∩ (◡𝐹 “ {𝑤}))
18	17	adantrl 714	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) = ∩ (◡𝐹 “ {𝑤}))
19	16, 18	eqeq12d 2742	. . . 4 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) = ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) ↔ ∩ (◡𝐹 “ {𝑣}) = ∩ (◡𝐹 “ {𝑤})))
20		fveq2 6893	. . . . . . 7 ⊢ (∩ (◡𝐹 “ {𝑣}) = ∩ (◡𝐹 “ {𝑤}) → (𝐹‘∩ (◡𝐹 “ {𝑣})) = (𝐹‘∩ (◡𝐹 “ {𝑤})))
21	20	adantl 480	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ ∩ (◡𝐹 “ {𝑣}) = ∩ (◡𝐹 “ {𝑤})) → (𝐹‘∩ (◡𝐹 “ {𝑣})) = (𝐹‘∩ (◡𝐹 “ {𝑤})))
22		cnvimass 6083	. . . . . . . . . . 11 ⊢ (◡𝐹 “ {𝑣}) ⊆ dom 𝐹
23	22, 4	sseqtri 4015	. . . . . . . . . 10 ⊢ (◡𝐹 “ {𝑣}) ⊆ On
24	8	sselda 3978	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → 𝑣 ∈ ran 𝐹)
25		inisegn0 6100	. . . . . . . . . . 11 ⊢ (𝑣 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝑣}) ≠ ∅)
26	24, 25	sylib 217	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (◡𝐹 “ {𝑣}) ≠ ∅)
27		onint 7791	. . . . . . . . . 10 ⊢ (((◡𝐹 “ {𝑣}) ⊆ On ∧ (◡𝐹 “ {𝑣}) ≠ ∅) → ∩ (◡𝐹 “ {𝑣}) ∈ (◡𝐹 “ {𝑣}))
28	23, 26, 27	sylancr 585	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → ∩ (◡𝐹 “ {𝑣}) ∈ (◡𝐹 “ {𝑣}))
29		fniniseg 7065	. . . . . . . . . . 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 495	. . . . . . . . 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 715	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝐹‘∩ (◡𝐹 “ {𝑣})) = 𝑣)
34	33	adantr 479	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ ∩ (◡𝐹 “ {𝑣}) = ∩ (◡𝐹 “ {𝑤})) → (𝐹‘∩ (◡𝐹 “ {𝑣})) = 𝑣)
35		cnvimass 6083	. . . . . . . . . . 11 ⊢ (◡𝐹 “ {𝑤}) ⊆ dom 𝐹
36	35, 4	sseqtri 4015	. . . . . . . . . 10 ⊢ (◡𝐹 “ {𝑤}) ⊆ On
37	8	sselda 3978	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ ran 𝐹)
38		inisegn0 6100	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝑤}) ≠ ∅)
39	37, 38	sylib 217	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (◡𝐹 “ {𝑤}) ≠ ∅)
40		onint 7791	. . . . . . . . . 10 ⊢ (((◡𝐹 “ {𝑤}) ⊆ On ∧ (◡𝐹 “ {𝑤}) ≠ ∅) → ∩ (◡𝐹 “ {𝑤}) ∈ (◡𝐹 “ {𝑤}))
41	36, 39, 40	sylancr 585	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ∩ (◡𝐹 “ {𝑤}) ∈ (◡𝐹 “ {𝑤}))
42		fniniseg 7065	. . . . . . . . . . 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 495	. . . . . . . . 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 714	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝐹‘∩ (◡𝐹 “ {𝑤})) = 𝑤)
47	46	adantr 479	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ ∩ (◡𝐹 “ {𝑣}) = ∩ (◡𝐹 “ {𝑤})) → (𝐹‘∩ (◡𝐹 “ {𝑤})) = 𝑤)
48	21, 34, 47	3eqtr3d 2774	. . . . 5 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ ∩ (◡𝐹 “ {𝑣}) = ∩ (◡𝐹 “ {𝑤})) → 𝑣 = 𝑤)
49	48	ex 411	. . . 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 3191	. 2 ⊢ (𝜑 → ∀𝑣 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) = ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) → 𝑣 = 𝑤))
52		dff13 7262	. 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 581	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})):𝐴–1-1→On)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1534   ∈ wcel 2099   ≠ wne 2930  ∀wral 3051  Vcvv 3462   ∖ cdif 3943   ⊆ wss 3946  ∅c0 4322  𝒫 cpw 4597  {csn 4623  ∩ cint 4946   ↦ cmpt 5228  ^◡ccnv 5673  dom cdm 5674  ran crn 5675   “ cima 5677  Oncon0 6368   Fn wfn 6541  ⟶wf 6542  –1-1→wf1 6543  ‘cfv 6546  recscrecs 8392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pr 5425  ax-un 7738

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-int 4947  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-tr 5263  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6304  df-ord 6371  df-on 6372  df-suc 6374  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-ov 7419  df-2nd 7996  df-frecs 8288  df-wrecs 8319  df-recs 8393

This theorem is referenced by:  dnwech  42746