Theorem bnj1385 35508

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1385.1	⊢ (𝜑 ↔ ∀𝑓 ∈ 𝐴 Fun 𝑓)
bnj1385.2	⊢ 𝐷 = (dom 𝑓 ∩ dom 𝑔)
bnj1385.3	⊢ (𝜓 ↔ (𝜑 ∧ ∀𝑓 ∈ 𝐴 ∀𝑔 ∈ 𝐴 (𝑓 ↾ 𝐷) = (𝑔 ↾ 𝐷)))
bnj1385.4	⊢ (𝑥 ∈ 𝐴 → ∀𝑓 𝑥 ∈ 𝐴)
bnj1385.5	⊢ (𝜑′ ↔ ∀ℎ ∈ 𝐴 Fun ℎ)
bnj1385.6	⊢ 𝐸 = (dom ℎ ∩ dom 𝑔)
bnj1385.7	⊢ (𝜓′ ↔ (𝜑′ ∧ ∀ℎ ∈ 𝐴 ∀𝑔 ∈ 𝐴 (ℎ ↾ 𝐸) = (𝑔 ↾ 𝐸)))

Assertion

Ref	Expression
bnj1385	⊢ (𝜓 → Fun ∪ 𝐴)

Distinct variable groups:   𝐴,𝑔,ℎ,𝑥   𝐷,ℎ   𝑓,𝐸   𝑓,𝑔,ℎ,𝑥   𝑔,𝜑′

Allowed substitution hints:   𝜑(𝑥, 𝑓, 𝑔, ℎ)   𝜓(𝑥, 𝑓, 𝑔, ℎ)   𝐴(𝑓)   𝐷(𝑥, 𝑓, 𝑔)   𝐸(𝑥, 𝑔, ℎ)   𝜑′(𝑥, 𝑓, ℎ)   𝜓′(𝑥, 𝑓, 𝑔, ℎ)

Proof of Theorem bnj1385

Step	Hyp	Ref	Expression
1		nfv 1946	. . . . . . 7 ⊢ Ⅎℎ(𝑓 ∈ 𝐴 → Fun 𝑓)
2		bnj1385.4	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → ∀𝑓 𝑥 ∈ 𝐴)
3	2	nfcii 2911	. . . . . . . . 9 ⊢ Ⅎ𝑓𝐴
4	3	nfcri 2914	. . . . . . . 8 ⊢ Ⅎ𝑓 ℎ ∈ 𝐴
5		nfv 1946	. . . . . . . 8 ⊢ Ⅎ𝑓Fun ℎ
6	4, 5	nfim 1928	. . . . . . 7 ⊢ Ⅎ𝑓(ℎ ∈ 𝐴 → Fun ℎ)
7		eleq1w 2843	. . . . . . . 8 ⊢ (𝑓 = ℎ → (𝑓 ∈ 𝐴 ↔ ℎ ∈ 𝐴))
8		funeq 6559	. . . . . . . 8 ⊢ (𝑓 = ℎ → (Fun 𝑓 ↔ Fun ℎ))
9	7, 8	imbi12d 347	. . . . . . 7 ⊢ (𝑓 = ℎ → ((𝑓 ∈ 𝐴 → Fun 𝑓) ↔ (ℎ ∈ 𝐴 → Fun ℎ)))
10	1, 6, 9	cbvalv1 2370	. . . . . 6 ⊢ (∀𝑓(𝑓 ∈ 𝐴 → Fun 𝑓) ↔ ∀ℎ(ℎ ∈ 𝐴 → Fun ℎ))
11		df-ral 3077	. . . . . 6 ⊢ (∀𝑓 ∈ 𝐴 Fun 𝑓 ↔ ∀𝑓(𝑓 ∈ 𝐴 → Fun 𝑓))
12		df-ral 3077	. . . . . 6 ⊢ (∀ℎ ∈ 𝐴 Fun ℎ ↔ ∀ℎ(ℎ ∈ 𝐴 → Fun ℎ))
13	10, 11, 12	3bitr4i 306	. . . . 5 ⊢ (∀𝑓 ∈ 𝐴 Fun 𝑓 ↔ ∀ℎ ∈ 𝐴 Fun ℎ)
14		bnj1385.1	. . . . 5 ⊢ (𝜑 ↔ ∀𝑓 ∈ 𝐴 Fun 𝑓)
15		bnj1385.5	. . . . 5 ⊢ (𝜑′ ↔ ∀ℎ ∈ 𝐴 Fun ℎ)
16	13, 14, 15	3bitr4i 306	. . . 4 ⊢ (𝜑 ↔ 𝜑′)
17		nfv 1946	. . . . . 6 ⊢ Ⅎℎ(𝑓 ∈ 𝐴 → ∀𝑔 ∈ 𝐴 (𝑓 ↾ 𝐷) = (𝑔 ↾ 𝐷))
18		nfv 1946	. . . . . . . 8 ⊢ Ⅎ𝑓(ℎ ↾ 𝐸) = (𝑔 ↾ 𝐸)
19	3, 18	nfralw 3309	. . . . . . 7 ⊢ Ⅎ𝑓∀𝑔 ∈ 𝐴 (ℎ ↾ 𝐸) = (𝑔 ↾ 𝐸)
20	4, 19	nfim 1928	. . . . . 6 ⊢ Ⅎ𝑓(ℎ ∈ 𝐴 → ∀𝑔 ∈ 𝐴 (ℎ ↾ 𝐸) = (𝑔 ↾ 𝐸))
21		dmeq 5884	. . . . . . . . . . . . 13 ⊢ (𝑓 = ℎ → dom 𝑓 = dom ℎ)
22	21	ineq1d 4116	. . . . . . . . . . . 12 ⊢ (𝑓 = ℎ → (dom 𝑓 ∩ dom 𝑔) = (dom ℎ ∩ dom 𝑔))
23		bnj1385.2	. . . . . . . . . . . 12 ⊢ 𝐷 = (dom 𝑓 ∩ dom 𝑔)
24		bnj1385.6	. . . . . . . . . . . 12 ⊢ 𝐸 = (dom ℎ ∩ dom 𝑔)
25	22, 23, 24	3eqtr4g 2820	. . . . . . . . . . 11 ⊢ (𝑓 = ℎ → 𝐷 = 𝐸)
26	25	reseq2d 5969	. . . . . . . . . 10 ⊢ (𝑓 = ℎ → (𝑓 ↾ 𝐷) = (𝑓 ↾ 𝐸))
27		reseq1 5963	. . . . . . . . . 10 ⊢ (𝑓 = ℎ → (𝑓 ↾ 𝐸) = (ℎ ↾ 𝐸))
28	26, 27	eqtrd 2795	. . . . . . . . 9 ⊢ (𝑓 = ℎ → (𝑓 ↾ 𝐷) = (ℎ ↾ 𝐸))
29	25	reseq2d 5969	. . . . . . . . 9 ⊢ (𝑓 = ℎ → (𝑔 ↾ 𝐷) = (𝑔 ↾ 𝐸))
30	28, 29	eqeq12d 2776	. . . . . . . 8 ⊢ (𝑓 = ℎ → ((𝑓 ↾ 𝐷) = (𝑔 ↾ 𝐷) ↔ (ℎ ↾ 𝐸) = (𝑔 ↾ 𝐸)))
31	30	ralbidv 3185	. . . . . . 7 ⊢ (𝑓 = ℎ → (∀𝑔 ∈ 𝐴 (𝑓 ↾ 𝐷) = (𝑔 ↾ 𝐷) ↔ ∀𝑔 ∈ 𝐴 (ℎ ↾ 𝐸) = (𝑔 ↾ 𝐸)))
32	7, 31	imbi12d 347	. . . . . 6 ⊢ (𝑓 = ℎ → ((𝑓 ∈ 𝐴 → ∀𝑔 ∈ 𝐴 (𝑓 ↾ 𝐷) = (𝑔 ↾ 𝐷)) ↔ (ℎ ∈ 𝐴 → ∀𝑔 ∈ 𝐴 (ℎ ↾ 𝐸) = (𝑔 ↾ 𝐸))))
33	17, 20, 32	cbvalv1 2370	. . . . 5 ⊢ (∀𝑓(𝑓 ∈ 𝐴 → ∀𝑔 ∈ 𝐴 (𝑓 ↾ 𝐷) = (𝑔 ↾ 𝐷)) ↔ ∀ℎ(ℎ ∈ 𝐴 → ∀𝑔 ∈ 𝐴 (ℎ ↾ 𝐸) = (𝑔 ↾ 𝐸)))
34		df-ral 3077	. . . . 5 ⊢ (∀𝑓 ∈ 𝐴 ∀𝑔 ∈ 𝐴 (𝑓 ↾ 𝐷) = (𝑔 ↾ 𝐷) ↔ ∀𝑓(𝑓 ∈ 𝐴 → ∀𝑔 ∈ 𝐴 (𝑓 ↾ 𝐷) = (𝑔 ↾ 𝐷)))
35		df-ral 3077	. . . . 5 ⊢ (∀ℎ ∈ 𝐴 ∀𝑔 ∈ 𝐴 (ℎ ↾ 𝐸) = (𝑔 ↾ 𝐸) ↔ ∀ℎ(ℎ ∈ 𝐴 → ∀𝑔 ∈ 𝐴 (ℎ ↾ 𝐸) = (𝑔 ↾ 𝐸)))
36	33, 34, 35	3bitr4i 306	. . . 4 ⊢ (∀𝑓 ∈ 𝐴 ∀𝑔 ∈ 𝐴 (𝑓 ↾ 𝐷) = (𝑔 ↾ 𝐷) ↔ ∀ℎ ∈ 𝐴 ∀𝑔 ∈ 𝐴 (ℎ ↾ 𝐸) = (𝑔 ↾ 𝐸))
37	16, 36	anbi12i 640	. . 3 ⊢ ((𝜑 ∧ ∀𝑓 ∈ 𝐴 ∀𝑔 ∈ 𝐴 (𝑓 ↾ 𝐷) = (𝑔 ↾ 𝐷)) ↔ (𝜑′ ∧ ∀ℎ ∈ 𝐴 ∀𝑔 ∈ 𝐴 (ℎ ↾ 𝐸) = (𝑔 ↾ 𝐸)))
38		bnj1385.3	. . 3 ⊢ (𝜓 ↔ (𝜑 ∧ ∀𝑓 ∈ 𝐴 ∀𝑔 ∈ 𝐴 (𝑓 ↾ 𝐷) = (𝑔 ↾ 𝐷)))
39		bnj1385.7	. . 3 ⊢ (𝜓′ ↔ (𝜑′ ∧ ∀ℎ ∈ 𝐴 ∀𝑔 ∈ 𝐴 (ℎ ↾ 𝐸) = (𝑔 ↾ 𝐸)))
40	37, 38, 39	3bitr4i 306	. 2 ⊢ (𝜓 ↔ 𝜓′)
41	15, 24, 39	bnj1383 35507	. 2 ⊢ (𝜓′ → Fun ∪ 𝐴)
42	40, 41	sylbi 220	1 ⊢ (𝜓 → Fun ∪ 𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1567   = wceq 1569   ∈ wcel 2144  ∀wral 3076   ∩ cin 4089  ∪ cuni 4866  dom cdm 5650   ↾ cres 5652  Fun wfun 6532

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-res 5662  df-iota 6494  df-fun 6540  df-fv 6546

This theorem is used by:  bnj1386  35509