Theorem bnj155 32261

Description: Technical lemma for bnj153 32262. 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
bnj155.1	⊢ (𝜓1 ↔ [𝑔 / 𝑓]𝜓′)
bnj155.2	⊢ (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))

Assertion

Ref	Expression
bnj155	⊢ (𝜓1 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝑔‘suc 𝑖) = ∪ 𝑦 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅)))

Distinct variable groups:   𝐴,𝑓   𝑅,𝑓   𝑓,𝑔,𝑖,𝑦

Allowed substitution hints:   𝐴(𝑦,𝑔,𝑖)   𝑅(𝑦,𝑔,𝑖)   𝜓′(𝑦,𝑓,𝑔,𝑖)   𝜓₁(𝑦,𝑓,𝑔,𝑖)

Proof of Theorem bnj155

Step	Hyp	Ref	Expression
1		bnj155.1	. 2 ⊢ (𝜓1 ↔ [𝑔 / 𝑓]𝜓′)
2		bnj155.2	. . 3 ⊢ (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
3	2	sbcbii 3776	. 2 ⊢ ([𝑔 / 𝑓]𝜓′ ↔ [𝑔 / 𝑓]∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
4		vex 3444	. . 3 ⊢ 𝑔 ∈ V
5		fveq1 6644	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑓‘suc 𝑖) = (𝑔‘suc 𝑖))
6		fveq1 6644	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑖) = (𝑔‘𝑖))
7	6	iuneq1d 4908	. . . . . 6 ⊢ (𝑓 = 𝑔 → ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅))
8	5, 7	eqeq12d 2814	. . . . 5 ⊢ (𝑓 = 𝑔 → ((𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝑔‘suc 𝑖) = ∪ 𝑦 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅)))
9	8	imbi2d 344	. . . 4 ⊢ (𝑓 = 𝑔 → ((suc 𝑖 ∈ 1o → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ (suc 𝑖 ∈ 1o → (𝑔‘suc 𝑖) = ∪ 𝑦 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅))))
10	9	ralbidv 3162	. . 3 ⊢ (𝑓 = 𝑔 → (∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝑔‘suc 𝑖) = ∪ 𝑦 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅))))
11	4, 10	sbcie 3760	. 2 ⊢ ([𝑔 / 𝑓]∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝑔‘suc 𝑖) = ∪ 𝑦 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅)))
12	1, 3, 11	3bitri 300	1 ⊢ (𝜓1 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝑔‘suc 𝑖) = ∪ 𝑦 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅)))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2111  ∀wral 3106  [wsbc 3720  ∪ ciun 4881  suc csuc 6161  ‘cfv 6324  ωcom 7560  1_oc1o 8078   predc-bnj14 32068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-in 3888  df-ss 3898  df-uni 4801  df-iun 4883  df-br 5031  df-iota 6283  df-fv 6332

This theorem is referenced by:  bnj153  32262