Theorem pwfseqlem2 10632

Description: Lemma for pwfseq 10637. (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 7407	. . 3 ⊢ (𝑎 = 𝑌 → (𝑎𝐹𝑠) = (𝑌𝐹𝑠))
2		2fveq3 6876	. . 3 ⊢ (𝑎 = 𝑌 → (𝐻‘(card‘𝑎)) = (𝐻‘(card‘𝑌)))
3	1, 2	eqeq12d 2781	. 2 ⊢ (𝑎 = 𝑌 → ((𝑎𝐹𝑠) = (𝐻‘(card‘𝑎)) ↔ (𝑌𝐹𝑠) = (𝐻‘(card‘𝑌))))
4		oveq2 7408	. . 3 ⊢ (𝑠 = 𝑅 → (𝑌𝐹𝑠) = (𝑌𝐹𝑅))
5	4	eqeq1d 2767	. 2 ⊢ (𝑠 = 𝑅 → ((𝑌𝐹𝑠) = (𝐻‘(card‘𝑌)) ↔ (𝑌𝐹𝑅) = (𝐻‘(card‘𝑌))))
6		nfcv 2927	. . 3 ⊢ Ⅎ𝑥𝑎
7		nfcv 2927	. . 3 ⊢ Ⅎ𝑟𝑎
8		nfcv 2927	. . 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 7480	. . . . . 6 ⊢ Ⅎ𝑥(𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥})))
11	9, 10	nfcxfr 2925	. . . . 5 ⊢ Ⅎ𝑥𝐹
12		nfcv 2927	. . . . 5 ⊢ Ⅎ𝑥𝑟
13	6, 11, 12	nfov 7430	. . . 4 ⊢ Ⅎ𝑥(𝑎𝐹𝑟)
14	13	nfeq1 2942	. . 3 ⊢ Ⅎ𝑥(𝑎𝐹𝑟) = (𝐻‘(card‘𝑎))
15		nfmpo2 7481	. . . . . 6 ⊢ Ⅎ𝑟(𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥})))
16	9, 15	nfcxfr 2925	. . . . 5 ⊢ Ⅎ𝑟𝐹
17	7, 16, 8	nfov 7430	. . . 4 ⊢ Ⅎ𝑟(𝑎𝐹𝑠)
18	17	nfeq1 2942	. . 3 ⊢ Ⅎ𝑟(𝑎𝐹𝑠) = (𝐻‘(card‘𝑎))
19		oveq1 7407	. . . 4 ⊢ (𝑥 = 𝑎 → (𝑥𝐹𝑟) = (𝑎𝐹𝑟))
20		2fveq3 6876	. . . 4 ⊢ (𝑥 = 𝑎 → (𝐻‘(card‘𝑥)) = (𝐻‘(card‘𝑎)))
21	19, 20	eqeq12d 2781	. . 3 ⊢ (𝑥 = 𝑎 → ((𝑥𝐹𝑟) = (𝐻‘(card‘𝑥)) ↔ (𝑎𝐹𝑟) = (𝐻‘(card‘𝑎))))
22		oveq2 7408	. . . 4 ⊢ (𝑟 = 𝑠 → (𝑎𝐹𝑟) = (𝑎𝐹𝑠))
23	22	eqeq1d 2767	. . 3 ⊢ (𝑟 = 𝑠 → ((𝑎𝐹𝑟) = (𝐻‘(card‘𝑎)) ↔ (𝑎𝐹𝑠) = (𝐻‘(card‘𝑎))))
24		vex 3461	. . . . . 6 ⊢ 𝑥 ∈ V
25		vex 3461	. . . . . 6 ⊢ 𝑟 ∈ V
26		fvex 6884	. . . . . . 7 ⊢ (𝐻‘(card‘𝑥)) ∈ V
27		fvex 6884	. . . . . . 7 ⊢ (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥}) ∈ V
28	26, 27	ifex 4534	. . . . . 6 ⊢ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥})) ∈ V
29	9	ovmpt4g 7547	. . . . . 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 1485	. . . . 5 ⊢ (𝑥𝐹𝑟) = if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥}))
31		iftrue 4489	. . . . 5 ⊢ (𝑥 ∈ Fin → if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥})) = (𝐻‘(card‘𝑥)))
32	30, 31	eqtrid 2812	. . . 4 ⊢ (𝑥 ∈ Fin → (𝑥𝐹𝑟) = (𝐻‘(card‘𝑥)))
33	32	adantr 485	. . 3 ⊢ ((𝑥 ∈ Fin ∧ 𝑟 ∈ 𝑉) → (𝑥𝐹𝑟) = (𝐻‘(card‘𝑥)))
34	6, 7, 8, 14, 18, 21, 23, 33	vtocl2gaf 3546	. 2 ⊢ ((𝑎 ∈ Fin ∧ 𝑠 ∈ 𝑉) → (𝑎𝐹𝑠) = (𝐻‘(card‘𝑎)))
35	3, 5, 34	vtocl2ga 3545	1 ⊢ ((𝑌 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑌𝐹𝑅) = (𝐻‘(card‘𝑌)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  {crab 3417  Vcvv 3457   ⊆ wss 3907  ifcif 4483  𝒫 cpw 4558  ∩ cint 4907  ∪ ciun 4951   class class class wbr 5104   We wwe 5603   × cxp 5649  ^◡ccnv 5650  ran crn 5652  –1-1→wf1 6522  –1-1-onto→wf1o 6524  ‘cfv 6525  (class class class)co 7400   ∈ cmpo 7402  ωcom 7850   ↑_m cmap 8812   ≼ cdom 8929  Fincfn 8931  cardccrd 9909

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5250  ax-nul 5260  ax-pr 5394

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5105  df-opab 5167  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6481  df-fun 6527  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405

This theorem is referenced by:  pwfseqlem4a  10634  pwfseqlem4  10635