Theorem bnj554 31304

Description: Technical lemma for bnj852 31326. 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
bnj554.19	⊢ (𝜂 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
bnj554.20	⊢ (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 = suc 𝑖))
bnj554.21	⊢ 𝐾 = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)
bnj554.22	⊢ 𝐿 = ∪ 𝑦 ∈ (𝐺‘𝑝) pred(𝑦, 𝐴, 𝑅)
bnj554.23	⊢ 𝐾 = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)
bnj554.24	⊢ 𝐿 = ∪ 𝑦 ∈ (𝐺‘𝑝) pred(𝑦, 𝐴, 𝑅)

Assertion

Ref	Expression
bnj554	⊢ ((𝜂 ∧ 𝜁) → ((𝐺‘𝑚) = 𝐿 ↔ (𝐺‘suc 𝑖) = 𝐾))

Distinct variable groups:   𝑦,𝐺   𝑦,𝑖   𝑦,𝑝

Allowed substitution hints:   𝜂(𝑦,𝑖,𝑚,𝑛,𝑝)   𝜁(𝑦,𝑖,𝑚,𝑛,𝑝)   𝐴(𝑦,𝑖,𝑚,𝑛,𝑝)   𝐷(𝑦,𝑖,𝑚,𝑛,𝑝)   𝑅(𝑦,𝑖,𝑚,𝑛,𝑝)   𝐺(𝑖,𝑚,𝑛,𝑝)   𝐾(𝑦,𝑖,𝑚,𝑛,𝑝)   𝐿(𝑦,𝑖,𝑚,𝑛,𝑝)

Proof of Theorem bnj554

Step	Hyp	Ref	Expression
1		bnj554.19	. . 3 ⊢ (𝜂 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
2	1	bnj1254 31215	. 2 ⊢ (𝜂 → 𝑚 = suc 𝑝)
3		bnj554.20	. . 3 ⊢ (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 = suc 𝑖))
4	3	simp3bi 1141	. 2 ⊢ (𝜁 → 𝑚 = suc 𝑖)
5		simpr 471	. . 3 ⊢ ((𝑚 = suc 𝑝 ∧ 𝑚 = suc 𝑖) → 𝑚 = suc 𝑖)
6		bnj551 31147	. . 3 ⊢ ((𝑚 = suc 𝑝 ∧ 𝑚 = suc 𝑖) → 𝑝 = 𝑖)
7		fveq2 6332	. . . 4 ⊢ (𝑚 = suc 𝑖 → (𝐺‘𝑚) = (𝐺‘suc 𝑖))
8		fveq2 6332	. . . . 5 ⊢ (𝑝 = 𝑖 → (𝐺‘𝑝) = (𝐺‘𝑖))
9		iuneq1 4668	. . . . . 6 ⊢ ((𝐺‘𝑝) = (𝐺‘𝑖) → ∪ 𝑦 ∈ (𝐺‘𝑝) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))
10		bnj554.24	. . . . . 6 ⊢ 𝐿 = ∪ 𝑦 ∈ (𝐺‘𝑝) pred(𝑦, 𝐴, 𝑅)
11		bnj554.23	. . . . . 6 ⊢ 𝐾 = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)
12	9, 10, 11	3eqtr4g 2830	. . . . 5 ⊢ ((𝐺‘𝑝) = (𝐺‘𝑖) → 𝐿 = 𝐾)
13	8, 12	syl 17	. . . 4 ⊢ (𝑝 = 𝑖 → 𝐿 = 𝐾)
14	7, 13	eqeqan12d 2787	. . 3 ⊢ ((𝑚 = suc 𝑖 ∧ 𝑝 = 𝑖) → ((𝐺‘𝑚) = 𝐿 ↔ (𝐺‘suc 𝑖) = 𝐾))
15	5, 6, 14	syl2anc 565	. 2 ⊢ ((𝑚 = suc 𝑝 ∧ 𝑚 = suc 𝑖) → ((𝐺‘𝑚) = 𝐿 ↔ (𝐺‘suc 𝑖) = 𝐾))
16	2, 4, 15	syl2an 575	1 ⊢ ((𝜂 ∧ 𝜁) → ((𝐺‘𝑚) = 𝐿 ↔ (𝐺‘suc 𝑖) = 𝐾))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145  ∪ ciun 4654  suc csuc 5868  ‘cfv 6031  ωcom 7211   ∧ w-bnj17 31089   predc-bnj14 31091

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-un 7095  ax-reg 8652

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-eprel 5162  df-fr 5208  df-suc 5872  df-iota 5994  df-fv 6039  df-bnj17 31090

This theorem is referenced by:  bnj558  31307