Theorem tz7.48-2 2995

Description: Proposition 7.48(2) of [TakeutiZaring] p. 51.

Hypothesis

Ref	Expression
tz7.48.1	⊢ F Fn On

Assertion

Ref	Expression
tz7.48-2	⊢ (∀x ∈ On (F ‘x) ∈ (A ∖ (F “ x)) → Fun ◡F)

Distinct variable group(s):   x,F   x,A

Proof of Theorem tz7.48-2

Step	Hyp	Ref	Expression
1		dmres 2584	. . . . . . . . . . . . . . . 16 ⊢ dom (F ↾ x) = (x ∩ dom F)
2	1	eleq2i 1153	. . . . . . . . . . . . . . 15 ⊢ (y ∈ dom (F ↾ x) ↔ y ∈ (x ∩ dom F))
3		elin 1635	. . . . . . . . . . . . . . 15 ⊢ (y ∈ (x ∩ dom F) ↔ (y ∈ x ∧ y ∈ dom F))
4	2, 3	bitr 151	. . . . . . . . . . . . . 14 ⊢ (y ∈ dom (F ↾ x) ↔ (y ∈ x ∧ y ∈ dom F))
5		tz7.48.1	. . . . . . . . . . . . . . . . 17 ⊢ F Fn On
6		fnfun 2721	. . . . . . . . . . . . . . . . 17 ⊢ (F Fn On → Fun F)
7	5, 6	ax-mp 6	. . . . . . . . . . . . . . . 16 ⊢ Fun F
8		funres 2697	. . . . . . . . . . . . . . . 16 ⊢ (Fun F → Fun (F ↾ x))
9	7, 8	ax-mp 6	. . . . . . . . . . . . . . 15 ⊢ Fun (F ↾ x)
10		fvrn 2888	. . . . . . . . . . . . . . 15 ⊢ ((Fun (F ↾ x) ∧ y ∈ dom (F ↾ x)) → ((F ↾ x) ‘y) ∈ ran (F ↾ x))
11	9, 10	mpan 518	. . . . . . . . . . . . . 14 ⊢ (y ∈ dom (F ↾ x) → ((F ↾ x) ‘y) ∈ ran (F ↾ x))
12	4, 11	sylbir 176	. . . . . . . . . . . . 13 ⊢ ((y ∈ x ∧ y ∈ dom F) → ((F ↾ x) ‘y) ∈ ran (F ↾ x))
13		fvres 2840	. . . . . . . . . . . . . . . 16 ⊢ (y ∈ x → ((F ↾ x) ‘y) = (F ‘y))
14	13	eleq1d 1155	. . . . . . . . . . . . . . 15 ⊢ (y ∈ x → (((F ↾ x) ‘y) ∈ ran (F ↾ x) ↔ (F ‘y) ∈ ran (F ↾ x)))
15		df-ima 2431	. . . . . . . . . . . . . . . 16 ⊢ (F “ x) = ran (F ↾ x)
16	15	eleq2i 1153	. . . . . . . . . . . . . . 15 ⊢ ((F ‘y) ∈ (F “ x) ↔ (F ‘y) ∈ ran (F ↾ x))
17	14, 16	syl6rbbr 417	. . . . . . . . . . . . . 14 ⊢ (y ∈ x → ((F ‘y) ∈ (F “ x) ↔ ((F ↾ x) ‘y) ∈ ran (F ↾ x)))
18	17	adantr 306	. . . . . . . . . . . . 13 ⊢ ((y ∈ x ∧ y ∈ dom F) → ((F ‘y) ∈ (F “ x) ↔ ((F ↾ x) ‘y) ∈ ran (F ↾ x)))
19	12, 18	mpbird 171	. . . . . . . . . . . 12 ⊢ ((y ∈ x ∧ y ∈ dom F) → (F ‘y) ∈ (F “ x))
20		eleq1a 1158	. . . . . . . . . . . . 13 ⊢ ((F ‘y) ∈ (F “ x) → ((F ‘x) = (F ‘y) → (F ‘x) ∈ (F “ x)))
21		eldifn 1592	. . . . . . . . . . . . 13 ⊢ ((F ‘x) ∈ (A ∖ (F “ x)) → ¬ (F ‘x) ∈ (F “ x))
22	20, 21	nsyli 106	. . . . . . . . . . . 12 ⊢ ((F ‘y) ∈ (F “ x) → ((F ‘x) ∈ (A ∖ (F “ x)) → ¬ (F ‘x) = (F ‘y)))
23	19, 22	syl 12	. . . . . . . . . . 11 ⊢ ((y ∈ x ∧ y ∈ dom F) → ((F ‘x) ∈ (A ∖ (F “ x)) → ¬ (F ‘x) = (F ‘y)))
24		fndm 2723	. . . . . . . . . . . . 13 ⊢ (F Fn On → dom F = On)
25	5, 24	ax-mp 6	. . . . . . . . . . . 12 ⊢ dom F = On
26	25	eleq2i 1153	. . . . . . . . . . 11 ⊢ (y ∈ dom F ↔ y ∈ On)
27	23, 26	sylan2br 348	. . . . . . . . . 10 ⊢ ((y ∈ x ∧ y ∈ On) → ((F ‘x) ∈ (A ∖ (F “ x)) → ¬ (F ‘x) = (F ‘y)))
28		pm3.26 256	. . . . . . . . . 10 ⊢ ((y ∈ x ∧ x ∈ On) → y ∈ x)
29		onelon 2223	. . . . . . . . . . 11 ⊢ ((x ∈ On ∧ y ∈ x) → y ∈ On)
30	29	ancoms 334	. . . . . . . . . 10 ⊢ ((y ∈ x ∧ x ∈ On) → y ∈ On)
31	27, 28, 30	sylanc 361	. . . . . . . . 9 ⊢ ((y ∈ x ∧ x ∈ On) → ((F ‘x) ∈ (A ∖ (F “ x)) → ¬ (F ‘x) = (F ‘y)))
32	31	exp 291	. . . . . . . 8 ⊢ (y ∈ x → (x ∈ On → ((F ‘x) ∈ (A ∖ (F “ x)) → ¬ (F ‘x) = (F ‘y))))
33	32	imp3a 279	. . . . . . 7 ⊢ (y ∈ x → ((x ∈ On ∧ (F ‘x) ∈ (A ∖ (F “ x))) → ¬ (F ‘x) = (F ‘y)))
34	33	com12 13	. . . . . 6 ⊢ ((x ∈ On ∧ (F ‘x) ∈ (A ∖ (F “ x))) → (y ∈ x → ¬ (F ‘x) = (F ‘y)))
35	34	r19.21aiv 1259	. . . . 5 ⊢ ((x ∈ On ∧ (F ‘x) ∈ (A ∖ (F “ x))) → ∀y ∈ x ¬ (F ‘x) = (F ‘y))
36	35	exp 291	. . . 4 ⊢ (x ∈ On → ((F ‘x) ∈ (A ∖ (F “ x)) → ∀y ∈ x ¬ (F ‘x) = (F ‘y)))
37	36	r19.20i 1253	. . 3 ⊢ (∀x ∈ On (F ‘x) ∈ (A ∖ (F “ x)) → ∀x ∈ On ∀y ∈ x ¬ (F ‘x) = (F ‘y))
38		ssid 1519	. . . 4 ⊢ On ⊆ On
39	5	tz7.48lem 2993	. . . 4 ⊢ ((On ⊆ On ∧ ∀x ∈ On ∀y ∈ x ¬ (F ‘x) = (F ‘y)) → Fun ◡(F ↾ On))
40	38, 39	mpan 518	. . 3 ⊢ (∀x ∈ On ∀y ∈ x ¬ (F ‘x) = (F ‘y) → Fun ◡(F ↾ On))
41	37, 40	syl 12	. 2 ⊢ (∀x ∈ On (F ‘x) ∈ (A ∖ (F “ x)) → Fun ◡(F ↾ On))
42		fnrel 2722	. . . . . 6 ⊢ (F Fn On → Rel F)
43	5, 42	ax-mp 6	. . . . 5 ⊢ Rel F
44	25, 38	eqsstr 1530	. . . . 5 ⊢ dom F ⊆ On
45		relssres 2596	. . . . 5 ⊢ ((Rel F ∧ dom F ⊆ On) → (F ↾ On) = F)
46	43, 44, 45	mp2an 520	. . . 4 ⊢ (F ↾ On) = F
47		cnveq 2513	. . . 4 ⊢ ((F ↾ On) = F → ◡(F ↾ On) = ◡F)
48	46, 47	ax-mp 6	. . 3 ⊢ ◡(F ↾ On) = ◡F
49		funeq 2683	. . 3 ⊢ (◡(F ↾ On) = ◡F → (Fun ◡(F ↾ On) ↔ Fun ◡F))
50	48, 49	ax-mp 6	. 2 ⊢ (Fun ◡(F ↾ On) ↔ Fun ◡F)
51	41, 50	sylib 173	1 ⊢ (∀x ∈ On (F ‘x) ∈ (A ∖ (F “ x)) → Fun ◡F)

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196   ∈ wel 803   = wceq 1091   ∈ wcel 1092  ∀wral 1201   ∖ cdif 1484   ∩ cin 1486   ⊆ wss 1487  Oncon0 2199  ^◡ccnv 2409  dom cdm 2410  ran crn 2411   ↾ cres 2412   “ cima 2413  Rel wrel 2415  Fun wfun 2416   Fn wfn 2417   ‘cfv 2422

This theorem is referenced by:  tz7.48-3 2996

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438

metamath.org