Theorem pwfseqlem2 10604

Description: Lemma for pwfseq 10609. (Contributed by Mario Carneiro, 18-Nov-2014.) (Revised by AV, 18-Sep-2021.)

Hypotheses

Ref	Expression
pwfseqlem4.g	⊢ (𝜑 → 𝐺:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
pwfseqlem4.x	⊢ (𝜑 → 𝑋 ⊆ 𝐴)
pwfseqlem4.h	⊢ (𝜑 → 𝐻:ω–1-1-onto→𝑋)
pwfseqlem4.ps	⊢ (𝜓 ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥))
pwfseqlem4.k	⊢ ((𝜑 ∧ 𝜓) → 𝐾:∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)–1-1→𝑥)
pwfseqlem4.d	⊢ 𝐷 = (𝐺‘{𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))})
pwfseqlem4.f	⊢ 𝐹 = (𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥})))

Assertion

Ref	Expression
pwfseqlem2	⊢ ((𝑌 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑌𝐹𝑅) = (𝐻‘(card‘𝑌)))

Distinct variable groups:   𝑛,𝑟,𝑤,𝑥,𝑧   𝐷,𝑛,𝑧   𝑤,𝐺   𝑤,𝐾   𝐻,𝑟,𝑥,𝑧   𝜑,𝑛,𝑟,𝑥,𝑧   𝜓,𝑛,𝑧   𝐴,𝑛,𝑟,𝑥,𝑧   𝑉,𝑟,𝑥

Allowed substitution hints:   𝜑(𝑤)   𝜓(𝑥,𝑤,𝑟)   𝐴(𝑤)   𝐷(𝑥,𝑤,𝑟)   𝑅(𝑥,𝑧,𝑤,𝑛,𝑟)   𝐹(𝑥,𝑧,𝑤,𝑛,𝑟)   𝐺(𝑥,𝑧,𝑛,𝑟)   𝐻(𝑤,𝑛)   𝐾(𝑥,𝑧,𝑛,𝑟)   𝑉(𝑧,𝑤,𝑛)   𝑋(𝑥,𝑧,𝑤,𝑛,𝑟)   𝑌(𝑥,𝑧,𝑤,𝑛,𝑟)

Proof of Theorem pwfseqlem2

Dummy variables 𝑎 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7369	. . 3 ⊢ (𝑎 = 𝑌 → (𝑎𝐹𝑠) = (𝑌𝐹𝑠))
2		2fveq3 6852	. . 3 ⊢ (𝑎 = 𝑌 → (𝐻‘(card‘𝑎)) = (𝐻‘(card‘𝑌)))
3	1, 2	eqeq12d 2747	. 2 ⊢ (𝑎 = 𝑌 → ((𝑎𝐹𝑠) = (𝐻‘(card‘𝑎)) ↔ (𝑌𝐹𝑠) = (𝐻‘(card‘𝑌))))
4		oveq2 7370	. . 3 ⊢ (𝑠 = 𝑅 → (𝑌𝐹𝑠) = (𝑌𝐹𝑅))
5	4	eqeq1d 2733	. 2 ⊢ (𝑠 = 𝑅 → ((𝑌𝐹𝑠) = (𝐻‘(card‘𝑌)) ↔ (𝑌𝐹𝑅) = (𝐻‘(card‘𝑌))))
6		nfcv 2902	. . 3 ⊢ Ⅎ𝑥𝑎
7		nfcv 2902	. . 3 ⊢ Ⅎ𝑟𝑎
8		nfcv 2902	. . 3 ⊢ Ⅎ𝑟𝑠
9		pwfseqlem4.f	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥})))
10		nfmpo1 7442	. . . . . 6 ⊢ Ⅎ𝑥(𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥})))
11	9, 10	nfcxfr 2900	. . . . 5 ⊢ Ⅎ𝑥𝐹
12		nfcv 2902	. . . . 5 ⊢ Ⅎ𝑥𝑟
13	6, 11, 12	nfov 7392	. . . 4 ⊢ Ⅎ𝑥(𝑎𝐹𝑟)
14	13	nfeq1 2917	. . 3 ⊢ Ⅎ𝑥(𝑎𝐹𝑟) = (𝐻‘(card‘𝑎))
15		nfmpo2 7443	. . . . . 6 ⊢ Ⅎ𝑟(𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥})))
16	9, 15	nfcxfr 2900	. . . . 5 ⊢ Ⅎ𝑟𝐹
17	7, 16, 8	nfov 7392	. . . 4 ⊢ Ⅎ𝑟(𝑎𝐹𝑠)
18	17	nfeq1 2917	. . 3 ⊢ Ⅎ𝑟(𝑎𝐹𝑠) = (𝐻‘(card‘𝑎))
19		oveq1 7369	. . . 4 ⊢ (𝑥 = 𝑎 → (𝑥𝐹𝑟) = (𝑎𝐹𝑟))
20		2fveq3 6852	. . . 4 ⊢ (𝑥 = 𝑎 → (𝐻‘(card‘𝑥)) = (𝐻‘(card‘𝑎)))
21	19, 20	eqeq12d 2747	. . 3 ⊢ (𝑥 = 𝑎 → ((𝑥𝐹𝑟) = (𝐻‘(card‘𝑥)) ↔ (𝑎𝐹𝑟) = (𝐻‘(card‘𝑎))))
22		oveq2 7370	. . . 4 ⊢ (𝑟 = 𝑠 → (𝑎𝐹𝑟) = (𝑎𝐹𝑠))
23	22	eqeq1d 2733	. . 3 ⊢ (𝑟 = 𝑠 → ((𝑎𝐹𝑟) = (𝐻‘(card‘𝑎)) ↔ (𝑎𝐹𝑠) = (𝐻‘(card‘𝑎))))
24		vex 3450	. . . . . 6 ⊢ 𝑥 ∈ V
25		vex 3450	. . . . . 6 ⊢ 𝑟 ∈ V
26		fvex 6860	. . . . . . 7 ⊢ (𝐻‘(card‘𝑥)) ∈ V
27		fvex 6860	. . . . . . 7 ⊢ (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥}) ∈ V
28	26, 27	ifex 4541	. . . . . 6 ⊢ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥})) ∈ V
29	9	ovmpt4g 7507	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑟 ∈ V ∧ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥})) ∈ V) → (𝑥𝐹𝑟) = if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥})))
30	24, 25, 28, 29	mp3an 1461	. . . . 5 ⊢ (𝑥𝐹𝑟) = if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥}))
31		iftrue 4497	. . . . 5 ⊢ (𝑥 ∈ Fin → if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥})) = (𝐻‘(card‘𝑥)))
32	30, 31	eqtrid 2783	. . . 4 ⊢ (𝑥 ∈ Fin → (𝑥𝐹𝑟) = (𝐻‘(card‘𝑥)))
33	32	adantr 481	. . 3 ⊢ ((𝑥 ∈ Fin ∧ 𝑟 ∈ 𝑉) → (𝑥𝐹𝑟) = (𝐻‘(card‘𝑥)))
34	6, 7, 8, 14, 18, 21, 23, 33	vtocl2gaf 3537	. 2 ⊢ ((𝑎 ∈ Fin ∧ 𝑠 ∈ 𝑉) → (𝑎𝐹𝑠) = (𝐻‘(card‘𝑎)))
35	3, 5, 34	vtocl2ga 3536	1 ⊢ ((𝑌 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑌𝐹𝑅) = (𝐻‘(card‘𝑌)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  {crab 3405  Vcvv 3446   ⊆ wss 3913  ifcif 4491  𝒫 cpw 4565  ∩ cint 4912  ∪ ciun 4959   class class class wbr 5110   We wwe 5592   × cxp 5636  ^◡ccnv 5637  ran crn 5639  –1-1→wf1 6498  –1-1-onto→wf1o 6500  ‘cfv 6501  (class class class)co 7362   ∈ cmpo 7364  ωcom 7807   ↑_m cmap 8772   ≼ cdom 8888  Fincfn 8890  cardccrd 9880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pr 5389

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3406  df-v 3448  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367

This theorem is referenced by:  pwfseqlem4a  10606  pwfseqlem4  10607