Theorem bnj155 30949

Description: Technical lemma for bnj153 30950. 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 3491	. 2 ⊢ ([𝑔 / 𝑓]𝜓′ ↔ [𝑔 / 𝑓]∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
4		vex 3203	. . 3 ⊢ 𝑔 ∈ V
5		fveq1 6190	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑓‘suc 𝑖) = (𝑔‘suc 𝑖))
6		fveq1 6190	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑖) = (𝑔‘𝑖))
7	6	iuneq1d 4545	. . . . . 6 ⊢ (𝑓 = 𝑔 → ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅))
8	5, 7	eqeq12d 2637	. . . . 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 330	. . . 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 2986	. . 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 3470	. 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 286	1 ⊢ (𝜓1 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑔‘suc 𝑖) = ∪ 𝑦 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅)))

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1483   ∈ wcel 1990  ∀wral 2912  [wsbc 3435  ∪ ciun 4520  suc csuc 5725  ‘cfv 5888  ωcom 7065  1_𝑜c1o 7553   predc-bnj14 30754

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-v 3202  df-sbc 3436  df-in 3581  df-ss 3588  df-uni 4437  df-iun 4522  df-br 4654  df-iota 5851  df-fv 5896

This theorem is referenced by:  bnj153  30950