Theorem bnj548 35294

Description: Technical lemma for bnj852 35318. 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
bnj548.1	⊢ (𝜏 ↔ (𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′))
bnj548.2	⊢ 𝐵 = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)
bnj548.3	⊢ 𝐾 = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)
bnj548.4	⊢ 𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})
bnj548.5	⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → 𝐺 Fn 𝑛)

Assertion

Ref	Expression
bnj548	⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚) → 𝐵 = 𝐾)

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

Allowed substitution hints:   𝜏(𝑦, 𝑓, 𝑖, 𝑚, 𝑛)   𝜎(𝑦, 𝑓, 𝑖, 𝑚, 𝑛)   𝐴(𝑦, 𝑓, 𝑖, 𝑚, 𝑛)   𝐵(𝑦, 𝑓, 𝑖, 𝑚, 𝑛)   𝐶(𝑦, 𝑓, 𝑖, 𝑚, 𝑛)   𝑅(𝑦, 𝑓, 𝑖, 𝑚, 𝑛)   𝐺(𝑓, 𝑖, 𝑚, 𝑛)   𝐾(𝑦, 𝑓, 𝑖, 𝑚, 𝑛)   𝜑′(𝑦, 𝑓, 𝑖, 𝑚, 𝑛)   𝜓′(𝑦, 𝑓, 𝑖, 𝑚, 𝑛)

Proof of Theorem bnj548

Step	Hyp	Ref	Expression
1		bnj548.5	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → 𝐺 Fn 𝑛)
2	1	fnfund 6636	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → Fun 𝐺)
3	2	adantr 485	. . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚) → Fun 𝐺)
4		bnj548.1	. . . . . . . 8 ⊢ (𝜏 ↔ (𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′))
5	4	simp1bi 1162	. . . . . . 7 ⊢ (𝜏 → 𝑓 Fn 𝑚)
6		fndm 6638	. . . . . . . 8 ⊢ (𝑓 Fn 𝑚 → dom 𝑓 = 𝑚)
7		eleq2 2851	. . . . . . . . 9 ⊢ (dom 𝑓 = 𝑚 → (𝑖 ∈ dom 𝑓 ↔ 𝑖 ∈ 𝑚))
8	7	biimpar 482	. . . . . . . 8 ⊢ ((dom 𝑓 = 𝑚 ∧ 𝑖 ∈ 𝑚) → 𝑖 ∈ dom 𝑓)
9	6, 8	sylan 591	. . . . . . 7 ⊢ ((𝑓 Fn 𝑚 ∧ 𝑖 ∈ 𝑚) → 𝑖 ∈ dom 𝑓)
10	5, 9	sylan 591	. . . . . 6 ⊢ ((𝜏 ∧ 𝑖 ∈ 𝑚) → 𝑖 ∈ dom 𝑓)
11	10	3ad2antl2 1204	. . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚) → 𝑖 ∈ dom 𝑓)
12	3, 11	jca 520	. . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚) → (Fun 𝐺 ∧ 𝑖 ∈ dom 𝑓))
13		bnj548.4	. . . . 5 ⊢ 𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})
14	13	bnj931 35168	. . . 4 ⊢ 𝑓 ⊆ 𝐺
15	12, 14	jctil 528	. . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚) → (𝑓 ⊆ 𝐺 ∧ (Fun 𝐺 ∧ 𝑖 ∈ dom 𝑓)))
16		3anan12 1111	. . 3 ⊢ ((Fun 𝐺 ∧ 𝑓 ⊆ 𝐺 ∧ 𝑖 ∈ dom 𝑓) ↔ (𝑓 ⊆ 𝐺 ∧ (Fun 𝐺 ∧ 𝑖 ∈ dom 𝑓)))
17	15, 16	sylibr 237	. 2 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚) → (Fun 𝐺 ∧ 𝑓 ⊆ 𝐺 ∧ 𝑖 ∈ dom 𝑓))
18		funssfv 6902	. 2 ⊢ ((Fun 𝐺 ∧ 𝑓 ⊆ 𝐺 ∧ 𝑖 ∈ dom 𝑓) → (𝐺‘𝑖) = (𝑓‘𝑖))
19		iuneq1 4972	. . . 4 ⊢ ((𝐺‘𝑖) = (𝑓‘𝑖) → ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))
20	19	eqcomd 2768	. . 3 ⊢ ((𝐺‘𝑖) = (𝑓‘𝑖) → ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))
21		bnj548.2	. . 3 ⊢ 𝐵 = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)
22		bnj548.3	. . 3 ⊢ 𝐾 = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)
23	20, 21, 22	3eqtr4g 2822	. 2 ⊢ ((𝐺‘𝑖) = (𝑓‘𝑖) → 𝐵 = 𝐾)
24	17, 18, 23	3syl 19	1 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚) → 𝐵 = 𝐾)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142   ∪ cun 3902   ⊆ wss 3904  {csn 4588  ⟨cop 4594  ∪ ciun 4955  dom cdm 5660  Fun wfun 6530   Fn wfn 6531  ‘cfv 6536   predc-bnj14 35086   FrSe w-bnj15 35090

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-res 5672  df-iota 6492  df-fun 6538  df-fn 6539  df-fv 6544

This theorem is used by:  bnj553  35295