Theorem bnj106 31245

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj106.1	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj106.2	⊢ 𝐹 ∈ V

Assertion

Ref	Expression
bnj106	⊢ ([𝐹 / 𝑓][1𝑜 / 𝑛]𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)))

Distinct variable groups:   𝐴,𝑓,𝑛   𝑓,𝐹,𝑖,𝑦   𝑅,𝑓,𝑛   𝑖,𝑛,𝑦

Allowed substitution hints:   𝜓(𝑦,𝑓,𝑖,𝑛)   𝐴(𝑦,𝑖)   𝑅(𝑦,𝑖)   𝐹(𝑛)

Proof of Theorem bnj106

Step	Hyp	Ref	Expression
1		bnj106.1	. . . 4 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
2		bnj105 31099	. . . 4 ⊢ 1𝑜 ∈ V
3	1, 2	bnj92 31239	. . 3 ⊢ ([1𝑜 / 𝑛]𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
4	3	sbcbii 3632	. 2 ⊢ ([𝐹 / 𝑓][1𝑜 / 𝑛]𝜓 ↔ [𝐹 / 𝑓]∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
5		bnj106.2	. . 3 ⊢ 𝐹 ∈ V
6		fveq1 6351	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘suc 𝑖) = (𝐹‘suc 𝑖))
7		fveq1 6351	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑖) = (𝐹‘𝑖))
8	7	bnj1113 31163	. . . . . 6 ⊢ (𝑓 = 𝐹 → ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅))
9	6, 8	eqeq12d 2775	. . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)))
10	9	imbi2d 329	. . . 4 ⊢ (𝑓 = 𝐹 → ((suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ (suc 𝑖 ∈ 1𝑜 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅))))
11	10	ralbidv 3124	. . 3 ⊢ (𝑓 = 𝐹 → (∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅))))
12	5, 11	sbcie 3611	. 2 ⊢ ([𝐹 / 𝑓]∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)))
13	4, 12	bitri 264	1 ⊢ ([𝐹 / 𝑓][1𝑜 / 𝑛]𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)))

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1632   ∈ wcel 2139  ∀wral 3050  Vcvv 3340  [wsbc 3576  ∪ ciun 4672  suc csuc 5886  ‘cfv 6049  ωcom 7230  1_𝑜c1o 7722   predc-bnj14 31063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-pw 4304  df-sn 4322  df-uni 4589  df-iun 4674  df-br 4805  df-suc 5890  df-iota 6012  df-fv 6057  df-1o 7729

This theorem is referenced by:  bnj126  31250