Theorem bnj929 32448

Description: Technical lemma for bnj69 32522. 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
bnj929.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj929.4	⊢ (𝜑′ ↔ [𝑝 / 𝑛]𝜑)
bnj929.7	⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑′)
bnj929.10	⊢ 𝐷 = (ω ∖ {∅})
bnj929.13	⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
bnj929.50	⊢ 𝐶 ∈ V

Assertion

Ref	Expression
bnj929	⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) → 𝜑″)

Distinct variable groups:   𝐴,𝑓,𝑛   𝑅,𝑓,𝑛   𝑓,𝑋,𝑛

Allowed substitution hints:   𝜑(𝑓,𝑛,𝑝)   𝐴(𝑝)   𝐶(𝑓,𝑛,𝑝)   𝐷(𝑓,𝑛,𝑝)   𝑅(𝑝)   𝐺(𝑓,𝑛,𝑝)   𝑋(𝑝)   𝜑′(𝑓,𝑛,𝑝)   𝜑″(𝑓,𝑛,𝑝)

Proof of Theorem bnj929

Step	Hyp	Ref	Expression
1		bnj645 32261	. 2 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) → 𝜑)
2		bnj334 32223	. . . . . . 7 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) ↔ (𝑓 Fn 𝑛 ∧ 𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝜑))
3		bnj257 32217	. . . . . . 7 ⊢ ((𝑓 Fn 𝑛 ∧ 𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝜑) ↔ (𝑓 Fn 𝑛 ∧ 𝑛 ∈ 𝐷 ∧ 𝜑 ∧ 𝑝 = suc 𝑛))
4	2, 3	bitri 278	. . . . . 6 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) ↔ (𝑓 Fn 𝑛 ∧ 𝑛 ∈ 𝐷 ∧ 𝜑 ∧ 𝑝 = suc 𝑛))
5		bnj345 32224	. . . . . 6 ⊢ ((𝑓 Fn 𝑛 ∧ 𝑛 ∈ 𝐷 ∧ 𝜑 ∧ 𝑝 = suc 𝑛) ↔ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝑛 ∈ 𝐷 ∧ 𝜑))
6		bnj253 32214	. . . . . 6 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝑛 ∈ 𝐷 ∧ 𝜑) ↔ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) ∧ 𝑛 ∈ 𝐷 ∧ 𝜑))
7	4, 5, 6	3bitri 300	. . . . 5 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) ↔ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) ∧ 𝑛 ∈ 𝐷 ∧ 𝜑))
8	7	simp1bi 1142	. . . 4 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) → (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛))
9		bnj929.13	. . . . . 6 ⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
10		bnj929.50	. . . . . 6 ⊢ 𝐶 ∈ V
11	9, 10	bnj927 32280	. . . . 5 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn 𝑝)
12	11	bnj930 32281	. . . 4 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → Fun 𝐺)
13	8, 12	syl 17	. . 3 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) → Fun 𝐺)
14	9	bnj931 32282	. . . 4 ⊢ 𝑓 ⊆ 𝐺
15	14	a1i 11	. . 3 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) → 𝑓 ⊆ 𝐺)
16		bnj268 32219	. . . . . 6 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝜑) ↔ (𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑))
17		bnj253 32214	. . . . . 6 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝜑) ↔ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛) ∧ 𝑝 = suc 𝑛 ∧ 𝜑))
18	16, 17	bitr3i 280	. . . . 5 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) ↔ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛) ∧ 𝑝 = suc 𝑛 ∧ 𝜑))
19	18	simp1bi 1142	. . . 4 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) → (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛))
20		fndm 6441	. . . . 5 ⊢ (𝑓 Fn 𝑛 → dom 𝑓 = 𝑛)
21		bnj929.10	. . . . . 6 ⊢ 𝐷 = (ω ∖ {∅})
22	21	bnj529 32252	. . . . 5 ⊢ (𝑛 ∈ 𝐷 → ∅ ∈ 𝑛)
23		eleq2 2840	. . . . . 6 ⊢ (dom 𝑓 = 𝑛 → (∅ ∈ dom 𝑓 ↔ ∅ ∈ 𝑛))
24	23	biimpar 481	. . . . 5 ⊢ ((dom 𝑓 = 𝑛 ∧ ∅ ∈ 𝑛) → ∅ ∈ dom 𝑓)
25	20, 22, 24	syl2anr 599	. . . 4 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛) → ∅ ∈ dom 𝑓)
26	19, 25	syl 17	. . 3 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) → ∅ ∈ dom 𝑓)
27	13, 15, 26	bnj1502 32360	. 2 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) → (𝐺‘∅) = (𝑓‘∅))
28		bnj929.1	. . 3 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
29		bnj929.4	. . 3 ⊢ (𝜑′ ↔ [𝑝 / 𝑛]𝜑)
30		bnj929.7	. . 3 ⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑′)
31	9	bnj918 32277	. . 3 ⊢ 𝐺 ∈ V
32	28, 29, 30, 31	bnj934 32447	. 2 ⊢ ((𝜑 ∧ (𝐺‘∅) = (𝑓‘∅)) → 𝜑″)
33	1, 27, 32	syl2anc 587	1 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) → 𝜑″)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  Vcvv 3409  [wsbc 3698   ∖ cdif 3857   ∪ cun 3858   ⊆ wss 3860  ∅c0 4227  {csn 4525  ⟨cop 4531  dom cdm 5528  suc csuc 6176  Fun wfun 6334   Fn wfn 6335  ‘cfv 6340  ωcom 7585   ∧ w-bnj17 32196   predc-bnj14 32198

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 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302  ax-un 7465  ax-reg 9102

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-res 5540  df-ord 6177  df-on 6178  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-fv 6348  df-om 7586  df-bnj17 32197

This theorem is referenced by:  bnj944  32450