Theorem bnj1014 31030

Description: Technical lemma for bnj69 31078. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1014.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj1014.2	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj1014.13	⊢ 𝐷 = (ω ∖ {∅})
bnj1014.14	⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}

Assertion

Ref	Expression
bnj1014	⊢ ((𝑔 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝑔) → (𝑔‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅))

Distinct variable groups:   𝐴,𝑓,𝑖,𝑛,𝑦   𝐷,𝑖   𝑅,𝑓,𝑖,𝑛,𝑦   𝑓,𝑋,𝑖,𝑛,𝑦   𝑓,𝑔,𝑖   𝑖,𝑗   𝜑,𝑖

Allowed substitution hints:   𝜑(𝑦,𝑓,𝑔,𝑗,𝑛)   𝜓(𝑦,𝑓,𝑔,𝑖,𝑗,𝑛)   𝐴(𝑔,𝑗)   𝐵(𝑦,𝑓,𝑔,𝑖,𝑗,𝑛)   𝐷(𝑦,𝑓,𝑔,𝑗,𝑛)   𝑅(𝑔,𝑗)   𝑋(𝑔,𝑗)

Proof of Theorem bnj1014

Step	Hyp	Ref	Expression
1		bnj1014.14	. . . . . . 7 ⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
2		nfcv 2764	. . . . . . . . 9 ⊢ Ⅎ𝑖𝐷
3		bnj1014.1	. . . . . . . . . . 11 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
4		bnj1014.2	. . . . . . . . . . 11 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
5	3, 4	bnj911 31002	. . . . . . . . . 10 ⊢ ((𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) → ∀𝑖(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
6	5	nf5i 2024	. . . . . . . . 9 ⊢ Ⅎ𝑖(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)
7	2, 6	nfrex 3007	. . . . . . . 8 ⊢ Ⅎ𝑖∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)
8	7	nfab 2769	. . . . . . 7 ⊢ Ⅎ𝑖{𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
9	1, 8	nfcxfr 2762	. . . . . 6 ⊢ Ⅎ𝑖𝐵
10	9	nfcri 2758	. . . . 5 ⊢ Ⅎ𝑖 𝑔 ∈ 𝐵
11		nfv 1843	. . . . 5 ⊢ Ⅎ𝑖 𝑗 ∈ dom 𝑔
12	10, 11	nfan 1828	. . . 4 ⊢ Ⅎ𝑖(𝑔 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝑔)
13		nfv 1843	. . . 4 ⊢ Ⅎ𝑖(𝑔‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅)
14	12, 13	nfim 1825	. . 3 ⊢ Ⅎ𝑖((𝑔 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝑔) → (𝑔‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅))
15	14	nf5ri 2065	. 2 ⊢ (((𝑔 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝑔) → (𝑔‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅)) → ∀𝑖((𝑔 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝑔) → (𝑔‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅)))
16		eleq1 2689	. . . . . 6 ⊢ (𝑗 = 𝑖 → (𝑗 ∈ dom 𝑔 ↔ 𝑖 ∈ dom 𝑔))
17	16	anbi2d 740	. . . . 5 ⊢ (𝑗 = 𝑖 → ((𝑔 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝑔) ↔ (𝑔 ∈ 𝐵 ∧ 𝑖 ∈ dom 𝑔)))
18		fveq2 6191	. . . . . 6 ⊢ (𝑗 = 𝑖 → (𝑔‘𝑗) = (𝑔‘𝑖))
19	18	sseq1d 3632	. . . . 5 ⊢ (𝑗 = 𝑖 → ((𝑔‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅) ↔ (𝑔‘𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅)))
20	17, 19	imbi12d 334	. . . 4 ⊢ (𝑗 = 𝑖 → (((𝑔 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝑔) → (𝑔‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅)) ↔ ((𝑔 ∈ 𝐵 ∧ 𝑖 ∈ dom 𝑔) → (𝑔‘𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))))
21	20	equcoms 1947	. . 3 ⊢ (𝑖 = 𝑗 → (((𝑔 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝑔) → (𝑔‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅)) ↔ ((𝑔 ∈ 𝐵 ∧ 𝑖 ∈ dom 𝑔) → (𝑔‘𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))))
22	1	bnj1317 30892	. . . . . . 7 ⊢ (𝑔 ∈ 𝐵 → ∀𝑓 𝑔 ∈ 𝐵)
23	22	nf5i 2024	. . . . . 6 ⊢ Ⅎ𝑓 𝑔 ∈ 𝐵
24		nfv 1843	. . . . . 6 ⊢ Ⅎ𝑓 𝑖 ∈ dom 𝑔
25	23, 24	nfan 1828	. . . . 5 ⊢ Ⅎ𝑓(𝑔 ∈ 𝐵 ∧ 𝑖 ∈ dom 𝑔)
26		nfv 1843	. . . . 5 ⊢ Ⅎ𝑓(𝑔‘𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅)
27	25, 26	nfim 1825	. . . 4 ⊢ Ⅎ𝑓((𝑔 ∈ 𝐵 ∧ 𝑖 ∈ dom 𝑔) → (𝑔‘𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))
28		eleq1 2689	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑓 ∈ 𝐵 ↔ 𝑔 ∈ 𝐵))
29		dmeq 5324	. . . . . . 7 ⊢ (𝑓 = 𝑔 → dom 𝑓 = dom 𝑔)
30	29	eleq2d 2687	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑖 ∈ dom 𝑓 ↔ 𝑖 ∈ dom 𝑔))
31	28, 30	anbi12d 747	. . . . 5 ⊢ (𝑓 = 𝑔 → ((𝑓 ∈ 𝐵 ∧ 𝑖 ∈ dom 𝑓) ↔ (𝑔 ∈ 𝐵 ∧ 𝑖 ∈ dom 𝑔)))
32		fveq1 6190	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑖) = (𝑔‘𝑖))
33	32	sseq1d 3632	. . . . 5 ⊢ (𝑓 = 𝑔 → ((𝑓‘𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅) ↔ (𝑔‘𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅)))
34	31, 33	imbi12d 334	. . . 4 ⊢ (𝑓 = 𝑔 → (((𝑓 ∈ 𝐵 ∧ 𝑖 ∈ dom 𝑓) → (𝑓‘𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅)) ↔ ((𝑔 ∈ 𝐵 ∧ 𝑖 ∈ dom 𝑔) → (𝑔‘𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))))
35		ssiun2 4563	. . . . 5 ⊢ (𝑖 ∈ dom 𝑓 → (𝑓‘𝑖) ⊆ ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖))
36		ssiun2 4563	. . . . . 6 ⊢ (𝑓 ∈ 𝐵 → ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖) ⊆ ∪ 𝑓 ∈ 𝐵 ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖))
37		bnj1014.13	. . . . . . 7 ⊢ 𝐷 = (ω ∖ {∅})
38	3, 4, 37, 1	bnj882 30996	. . . . . 6 ⊢ trCl(𝑋, 𝐴, 𝑅) = ∪ 𝑓 ∈ 𝐵 ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖)
39	36, 38	syl6sseqr 3652	. . . . 5 ⊢ (𝑓 ∈ 𝐵 → ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))
40	35, 39	sylan9ssr 3617	. . . 4 ⊢ ((𝑓 ∈ 𝐵 ∧ 𝑖 ∈ dom 𝑓) → (𝑓‘𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))
41	27, 34, 40	chvar 2262	. . 3 ⊢ ((𝑔 ∈ 𝐵 ∧ 𝑖 ∈ dom 𝑔) → (𝑔‘𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))
42	21, 41	spei 2261	. 2 ⊢ ∃𝑖((𝑔 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝑔) → (𝑔‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅))
43	15, 42	bnj1131 30858	1 ⊢ ((𝑔 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝑔) → (𝑔‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  {cab 2608  ∀wral 2912  ∃wrex 2913   ∖ cdif 3571   ⊆ wss 3574  ∅c0 3915  {csn 4177  ∪ ciun 4520  dom cdm 5114  suc csuc 5725   Fn wfn 5883  ‘cfv 5888  ωcom 7065   predc-bnj14 30754   trClc-bnj18 30760

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-dm 5124  df-iota 5851  df-fv 5896  df-bnj18 30761

This theorem is referenced by:  bnj1015  31031