Theorem bnj1148 33082

Description: Property of pred. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Assertion

Ref	Expression
bnj1148	⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → pred(𝑋, 𝐴, 𝑅) ∈ V)

Proof of Theorem bnj1148

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elisset 2819	. . . . 5 ⊢ (𝑋 ∈ 𝐴 → ∃𝑥 𝑥 = 𝑋)
2	1	adantl 482	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∃𝑥 𝑥 = 𝑋)
3		bnj93 32949	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V)
4		eleq1 2825	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝐴 ↔ 𝑋 ∈ 𝐴))
5	4	anbi2d 629	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴)))
6		bnj602 33001	. . . . . . 7 ⊢ (𝑥 = 𝑋 → pred(𝑥, 𝐴, 𝑅) = pred(𝑋, 𝐴, 𝑅))
7	6	eleq1d 2822	. . . . . 6 ⊢ (𝑥 = 𝑋 → ( pred(𝑥, 𝐴, 𝑅) ∈ V ↔ pred(𝑋, 𝐴, 𝑅) ∈ V))
8	5, 7	imbi12d 344	. . . . 5 ⊢ (𝑥 = 𝑋 → (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → pred(𝑋, 𝐴, 𝑅) ∈ V)))
9	3, 8	mpbii 232	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → pred(𝑋, 𝐴, 𝑅) ∈ V))
10	2, 9	bnj593 32831	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∃𝑥((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → pred(𝑋, 𝐴, 𝑅) ∈ V))
11	10	bnj937 32857	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → pred(𝑋, 𝐴, 𝑅) ∈ V))
12	11	pm2.43i 52	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → pred(𝑋, 𝐴, 𝑅) ∈ V)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1540  ∃wex 1780   ∈ wcel 2105  Vcvv 3441   predc-bnj14 32773   FrSe w-bnj15 32777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-12 2170  ax-ext 2708

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3063  df-rab 3405  df-v 3443  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-br 5086  df-bnj14 32774  df-bnj13 32776  df-bnj15 32778

This theorem is referenced by:  bnj1136  33083  bnj1413  33121