Theorem dnnumch3lem 39639

Description: Value of the ordinal injection 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
dnnumch3lem	⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) = ∩ (◡𝐹 “ {𝑤}))

Distinct variable groups:   𝑤,𝐹,𝑥,𝑦   𝑤,𝐺,𝑥,𝑦,𝑧   𝑤,𝐴,𝑥,𝑦,𝑧   𝜑,𝑥,𝑤

Allowed substitution hints:   𝜑(𝑦,𝑧)   𝐹(𝑧)   𝑉(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem dnnumch3lem

Step	Hyp	Ref	Expression
1		eqid 2821	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})) = (𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))
2		sneq 4570	. . . 4 ⊢ (𝑥 = 𝑤 → {𝑥} = {𝑤})
3	2	imaeq2d 5923	. . 3 ⊢ (𝑥 = 𝑤 → (◡𝐹 “ {𝑥}) = (◡𝐹 “ {𝑤}))
4	3	inteqd 4873	. 2 ⊢ (𝑥 = 𝑤 → ∩ (◡𝐹 “ {𝑥}) = ∩ (◡𝐹 “ {𝑤}))
5		simpr 487	. 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ 𝐴)
6		cnvimass 5943	. . . 4 ⊢ (◡𝐹 “ {𝑤}) ⊆ dom 𝐹
7		dnnumch.f	. . . . . 6 ⊢ 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))
8	7	tfr1 8027	. . . . 5 ⊢ 𝐹 Fn On
9		fndm 6449	. . . . 5 ⊢ (𝐹 Fn On → dom 𝐹 = On)
10	8, 9	ax-mp 5	. . . 4 ⊢ dom 𝐹 = On
11	6, 10	sseqtri 4002	. . 3 ⊢ (◡𝐹 “ {𝑤}) ⊆ On
12		dnnumch.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
13		dnnumch.g	. . . . . 6 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺‘𝑦) ∈ 𝑦))
14	7, 12, 13	dnnumch2 39638	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ran 𝐹)
15	14	sselda 3966	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ ran 𝐹)
16		inisegn0 5955	. . . 4 ⊢ (𝑤 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝑤}) ≠ ∅)
17	15, 16	sylib 220	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (◡𝐹 “ {𝑤}) ≠ ∅)
18		oninton 7509	. . 3 ⊢ (((◡𝐹 “ {𝑤}) ⊆ On ∧ (◡𝐹 “ {𝑤}) ≠ ∅) → ∩ (◡𝐹 “ {𝑤}) ∈ On)
19	11, 17, 18	sylancr 589	. 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ∩ (◡𝐹 “ {𝑤}) ∈ On)
20	1, 4, 5, 19	fvmptd3 6785	1 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) = ∩ (◡𝐹 “ {𝑤}))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110   ≠ wne 3016  ∀wral 3138  Vcvv 3494   ∖ cdif 3932   ⊆ wss 3935  ∅c0 4290  𝒫 cpw 4538  {csn 4560  ∩ cint 4868   ↦ cmpt 5138  ^◡ccnv 5548  dom cdm 5549  ran crn 5550   “ cima 5552  Oncon0 6185   Fn wfn 6344  ‘cfv 6349  recscrecs 8001

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4869  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-wrecs 7941  df-recs 8002

This theorem is referenced by:  dnnumch3  39640  dnwech  39641