Theorem bnj548 32544

Description: Technical lemma for bnj852 32568. 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	bnj930 32416	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → Fun 𝐺)
3	2	adantr 484	. . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚) → Fun 𝐺)
4		bnj548.1	. . . . . . . 8 ⊢ (𝜏 ↔ (𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′))
5	4	simp1bi 1147	. . . . . . 7 ⊢ (𝜏 → 𝑓 Fn 𝑚)
6		fndm 6459	. . . . . . . 8 ⊢ (𝑓 Fn 𝑚 → dom 𝑓 = 𝑚)
7		eleq2 2819	. . . . . . . . 9 ⊢ (dom 𝑓 = 𝑚 → (𝑖 ∈ dom 𝑓 ↔ 𝑖 ∈ 𝑚))
8	7	biimpar 481	. . . . . . . 8 ⊢ ((dom 𝑓 = 𝑚 ∧ 𝑖 ∈ 𝑚) → 𝑖 ∈ dom 𝑓)
9	6, 8	sylan 583	. . . . . . 7 ⊢ ((𝑓 Fn 𝑚 ∧ 𝑖 ∈ 𝑚) → 𝑖 ∈ dom 𝑓)
10	5, 9	sylan 583	. . . . . 6 ⊢ ((𝜏 ∧ 𝑖 ∈ 𝑚) → 𝑖 ∈ dom 𝑓)
11	10	3ad2antl2 1188	. . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚) → 𝑖 ∈ dom 𝑓)
12	3, 11	jca 515	. . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚) → (Fun 𝐺 ∧ 𝑖 ∈ dom 𝑓))
13		bnj548.4	. . . . 5 ⊢ 𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})
14	13	bnj931 32417	. . . 4 ⊢ 𝑓 ⊆ 𝐺
15	12, 14	jctil 523	. . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚) → (𝑓 ⊆ 𝐺 ∧ (Fun 𝐺 ∧ 𝑖 ∈ dom 𝑓)))
16		3anan12 1098	. . 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 6716	. 2 ⊢ ((Fun 𝐺 ∧ 𝑓 ⊆ 𝐺 ∧ 𝑖 ∈ dom 𝑓) → (𝐺‘𝑖) = (𝑓‘𝑖))
19		iuneq1 4906	. . . 4 ⊢ ((𝐺‘𝑖) = (𝑓‘𝑖) → ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))
20	19	eqcomd 2742	. . 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 2796	. 2 ⊢ ((𝐺‘𝑖) = (𝑓‘𝑖) → 𝐵 = 𝐾)
24	17, 18, 23	3syl 18	1 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚) → 𝐵 = 𝐾)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2112   ∪ cun 3851   ⊆ wss 3853  {csn 4527  ⟨cop 4533  ∪ ciun 4890  dom cdm 5536  Fun wfun 6352   Fn wfn 6353  ‘cfv 6358   predc-bnj14 32333   FrSe w-bnj15 32337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pr 5307

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-iun 4892  df-br 5040  df-opab 5102  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-res 5548  df-iota 6316  df-fun 6360  df-fn 6361  df-fv 6366

This theorem is referenced by:  bnj553  32545