Theorem bnj1015 34976

Description: Technical lemma for bnj69 35024. 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
bnj1015.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj1015.2	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj1015.13	⊢ 𝐷 = (ω ∖ {∅})
bnj1015.14	⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
bnj1015.15	⊢ 𝐺 ∈ 𝑉
bnj1015.16	⊢ 𝐽 ∈ 𝑉

Assertion

Ref	Expression
bnj1015	⊢ ((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   𝜑,𝑖

Allowed substitution hints:   𝜑(𝑦,𝑓,𝑛)   𝜓(𝑦,𝑓,𝑖,𝑛)   𝐵(𝑦,𝑓,𝑖,𝑛)   𝐷(𝑦,𝑓,𝑛)   𝐺(𝑦,𝑓,𝑖,𝑛)   𝐽(𝑦,𝑓,𝑖,𝑛)   𝑉(𝑦,𝑓,𝑖,𝑛)

Proof of Theorem bnj1015

Dummy variables 𝑔 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj1015.16	. . 3 ⊢ 𝐽 ∈ 𝑉
2	1	elexi 3503	. 2 ⊢ 𝐽 ∈ V
3		eleq1 2829	. . . 4 ⊢ (𝑗 = 𝐽 → (𝑗 ∈ dom 𝐺 ↔ 𝐽 ∈ dom 𝐺))
4	3	anbi2d 630	. . 3 ⊢ (𝑗 = 𝐽 → ((𝐺 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝐺) ↔ (𝐺 ∈ 𝐵 ∧ 𝐽 ∈ dom 𝐺)))
5		fveq2 6906	. . . 4 ⊢ (𝑗 = 𝐽 → (𝐺‘𝑗) = (𝐺‘𝐽))
6	5	sseq1d 4015	. . 3 ⊢ (𝑗 = 𝐽 → ((𝐺‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅) ↔ (𝐺‘𝐽) ⊆ trCl(𝑋, 𝐴, 𝑅)))
7	4, 6	imbi12d 344	. 2 ⊢ (𝑗 = 𝐽 → (((𝐺 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝐺) → (𝐺‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅)) ↔ ((𝐺 ∈ 𝐵 ∧ 𝐽 ∈ dom 𝐺) → (𝐺‘𝐽) ⊆ trCl(𝑋, 𝐴, 𝑅))))
8		bnj1015.15	. . . 4 ⊢ 𝐺 ∈ 𝑉
9	8	elexi 3503	. . 3 ⊢ 𝐺 ∈ V
10		eleq1 2829	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑔 ∈ 𝐵 ↔ 𝐺 ∈ 𝐵))
11		dmeq 5914	. . . . . 6 ⊢ (𝑔 = 𝐺 → dom 𝑔 = dom 𝐺)
12	11	eleq2d 2827	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑗 ∈ dom 𝑔 ↔ 𝑗 ∈ dom 𝐺))
13	10, 12	anbi12d 632	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝑔 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝑔) ↔ (𝐺 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝐺)))
14		fveq1 6905	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑔‘𝑗) = (𝐺‘𝑗))
15	14	sseq1d 4015	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝑔‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅) ↔ (𝐺‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅)))
16	13, 15	imbi12d 344	. . 3 ⊢ (𝑔 = 𝐺 → (((𝑔 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝑔) → (𝑔‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅)) ↔ ((𝐺 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝐺) → (𝐺‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅))))
17		bnj1015.1	. . . 4 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
18		bnj1015.2	. . . 4 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
19		bnj1015.13	. . . 4 ⊢ 𝐷 = (ω ∖ {∅})
20		bnj1015.14	. . . 4 ⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
21	17, 18, 19, 20	bnj1014 34975	. . 3 ⊢ ((𝑔 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝑔) → (𝑔‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅))
22	9, 16, 21	vtocl 3558	. 2 ⊢ ((𝐺 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝐺) → (𝐺‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅))
23	2, 7, 22	vtocl 3558	1 ⊢ ((𝐺 ∈ 𝐵 ∧ 𝐽 ∈ dom 𝐺) → (𝐺‘𝐽) ⊆ trCl(𝑋, 𝐴, 𝑅))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  {cab 2714  ∀wral 3061  ∃wrex 3070   ∖ cdif 3948   ⊆ wss 3951  ∅c0 4333  {csn 4626  ∪ ciun 4991  dom cdm 5685  suc csuc 6386   Fn wfn 6556  ‘cfv 6561  ωcom 7887   predc-bnj14 34702   trClc-bnj18 34708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-dm 5695  df-iota 6514  df-fv 6569  df-bnj18 34709

This theorem is referenced by:  bnj1018g  34977  bnj1018  34978