Theorem bnj1385 34988

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 1915	. . . . . . 7 ⊢ Ⅎℎ(𝑓 ∈ 𝐴 → Fun 𝑓)
2		bnj1385.4	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → ∀𝑓 𝑥 ∈ 𝐴)
3	2	nfcii 2887	. . . . . . . . 9 ⊢ Ⅎ𝑓𝐴
4	3	nfcri 2890	. . . . . . . 8 ⊢ Ⅎ𝑓 ℎ ∈ 𝐴
5		nfv 1915	. . . . . . . 8 ⊢ Ⅎ𝑓Fun ℎ
6	4, 5	nfim 1897	. . . . . . 7 ⊢ Ⅎ𝑓(ℎ ∈ 𝐴 → Fun ℎ)
7		eleq1w 2819	. . . . . . . 8 ⊢ (𝑓 = ℎ → (𝑓 ∈ 𝐴 ↔ ℎ ∈ 𝐴))
8		funeq 6512	. . . . . . . 8 ⊢ (𝑓 = ℎ → (Fun 𝑓 ↔ Fun ℎ))
9	7, 8	imbi12d 344	. . . . . . 7 ⊢ (𝑓 = ℎ → ((𝑓 ∈ 𝐴 → Fun 𝑓) ↔ (ℎ ∈ 𝐴 → Fun ℎ)))
10	1, 6, 9	cbvalv1 2345	. . . . . 6 ⊢ (∀𝑓(𝑓 ∈ 𝐴 → Fun 𝑓) ↔ ∀ℎ(ℎ ∈ 𝐴 → Fun ℎ))
11		df-ral 3052	. . . . . 6 ⊢ (∀𝑓 ∈ 𝐴 Fun 𝑓 ↔ ∀𝑓(𝑓 ∈ 𝐴 → Fun 𝑓))
12		df-ral 3052	. . . . . 6 ⊢ (∀ℎ ∈ 𝐴 Fun ℎ ↔ ∀ℎ(ℎ ∈ 𝐴 → Fun ℎ))
13	10, 11, 12	3bitr4i 303	. . . . 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 303	. . . 4 ⊢ (𝜑 ↔ 𝜑′)
17		nfv 1915	. . . . . 6 ⊢ Ⅎℎ(𝑓 ∈ 𝐴 → ∀𝑔 ∈ 𝐴 (𝑓 ↾ 𝐷) = (𝑔 ↾ 𝐷))
18		nfv 1915	. . . . . . . 8 ⊢ Ⅎ𝑓(ℎ ↾ 𝐸) = (𝑔 ↾ 𝐸)
19	3, 18	nfralw 3283	. . . . . . 7 ⊢ Ⅎ𝑓∀𝑔 ∈ 𝐴 (ℎ ↾ 𝐸) = (𝑔 ↾ 𝐸)
20	4, 19	nfim 1897	. . . . . 6 ⊢ Ⅎ𝑓(ℎ ∈ 𝐴 → ∀𝑔 ∈ 𝐴 (ℎ ↾ 𝐸) = (𝑔 ↾ 𝐸))
21		dmeq 5852	. . . . . . . . . . . . 13 ⊢ (𝑓 = ℎ → dom 𝑓 = dom ℎ)
22	21	ineq1d 4171	. . . . . . . . . . . 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 2796	. . . . . . . . . . 11 ⊢ (𝑓 = ℎ → 𝐷 = 𝐸)
26	25	reseq2d 5938	. . . . . . . . . 10 ⊢ (𝑓 = ℎ → (𝑓 ↾ 𝐷) = (𝑓 ↾ 𝐸))
27		reseq1 5932	. . . . . . . . . 10 ⊢ (𝑓 = ℎ → (𝑓 ↾ 𝐸) = (ℎ ↾ 𝐸))
28	26, 27	eqtrd 2771	. . . . . . . . 9 ⊢ (𝑓 = ℎ → (𝑓 ↾ 𝐷) = (ℎ ↾ 𝐸))
29	25	reseq2d 5938	. . . . . . . . 9 ⊢ (𝑓 = ℎ → (𝑔 ↾ 𝐷) = (𝑔 ↾ 𝐸))
30	28, 29	eqeq12d 2752	. . . . . . . 8 ⊢ (𝑓 = ℎ → ((𝑓 ↾ 𝐷) = (𝑔 ↾ 𝐷) ↔ (ℎ ↾ 𝐸) = (𝑔 ↾ 𝐸)))
31	30	ralbidv 3159	. . . . . . 7 ⊢ (𝑓 = ℎ → (∀𝑔 ∈ 𝐴 (𝑓 ↾ 𝐷) = (𝑔 ↾ 𝐷) ↔ ∀𝑔 ∈ 𝐴 (ℎ ↾ 𝐸) = (𝑔 ↾ 𝐸)))
32	7, 31	imbi12d 344	. . . . . 6 ⊢ (𝑓 = ℎ → ((𝑓 ∈ 𝐴 → ∀𝑔 ∈ 𝐴 (𝑓 ↾ 𝐷) = (𝑔 ↾ 𝐷)) ↔ (ℎ ∈ 𝐴 → ∀𝑔 ∈ 𝐴 (ℎ ↾ 𝐸) = (𝑔 ↾ 𝐸))))
33	17, 20, 32	cbvalv1 2345	. . . . 5 ⊢ (∀𝑓(𝑓 ∈ 𝐴 → ∀𝑔 ∈ 𝐴 (𝑓 ↾ 𝐷) = (𝑔 ↾ 𝐷)) ↔ ∀ℎ(ℎ ∈ 𝐴 → ∀𝑔 ∈ 𝐴 (ℎ ↾ 𝐸) = (𝑔 ↾ 𝐸)))
34		df-ral 3052	. . . . 5 ⊢ (∀𝑓 ∈ 𝐴 ∀𝑔 ∈ 𝐴 (𝑓 ↾ 𝐷) = (𝑔 ↾ 𝐷) ↔ ∀𝑓(𝑓 ∈ 𝐴 → ∀𝑔 ∈ 𝐴 (𝑓 ↾ 𝐷) = (𝑔 ↾ 𝐷)))
35		df-ral 3052	. . . . 5 ⊢ (∀ℎ ∈ 𝐴 ∀𝑔 ∈ 𝐴 (ℎ ↾ 𝐸) = (𝑔 ↾ 𝐸) ↔ ∀ℎ(ℎ ∈ 𝐴 → ∀𝑔 ∈ 𝐴 (ℎ ↾ 𝐸) = (𝑔 ↾ 𝐸)))
36	33, 34, 35	3bitr4i 303	. . . 4 ⊢ (∀𝑓 ∈ 𝐴 ∀𝑔 ∈ 𝐴 (𝑓 ↾ 𝐷) = (𝑔 ↾ 𝐷) ↔ ∀ℎ ∈ 𝐴 ∀𝑔 ∈ 𝐴 (ℎ ↾ 𝐸) = (𝑔 ↾ 𝐸))
37	16, 36	anbi12i 628	. . 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 303	. 2 ⊢ (𝜓 ↔ 𝜓′)
41	15, 24, 39	bnj1383 34987	. 2 ⊢ (𝜓′ → Fun ∪ 𝐴)
42	40, 41	sylbi 217	1 ⊢ (𝜓 → Fun ∪ 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1539   = wceq 1541   ∈ wcel 2113  ∀wral 3051   ∩ cin 3900  ∪ cuni 4863  dom cdm 5624   ↾ cres 5626  Fun wfun 6486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-res 5636  df-iota 6448  df-fun 6494  df-fv 6500

This theorem is referenced by:  bnj1386  34989