Theorem tz7.48-3 2996

Description: Proposition 7.48(3) of [TakeutiZaring] p. 51.

Hypothesis

Ref	Expression
tz7.48.1	⊢ F Fn On

Assertion

Ref	Expression
tz7.48-3	⊢ (∀x ∈ On (F ‘x) ∈ (A ∖ (F “ x)) → ¬ A ∈ V)

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

Proof of Theorem tz7.48-3

Step	Hyp	Ref	Expression
1		onprc 2240	. . . 4 ⊢ ¬ On ∈ V
2		tz7.48.1	. . . . . 6 ⊢ F Fn On
3		fndm 2723	. . . . . 6 ⊢ (F Fn On → dom F = On)
4	2, 3	ax-mp 6	. . . . 5 ⊢ dom F = On
5	4	eleq1i 1152	. . . 4 ⊢ (dom F ∈ V ↔ On ∈ V)
6	1, 5	mtbir 167	. . 3 ⊢ ¬ dom F ∈ V
7	2	tz7.48-2 2995	. . . 4 ⊢ (∀x ∈ On (F ‘x) ∈ (A ∖ (F “ x)) → Fun ◡F)
8		funrnex 2743	. . . . . 6 ⊢ (dom ◡F ∈ V → (Fun ◡F → ran ◡F ∈ V))
9	8	com12 13	. . . . 5 ⊢ (Fun ◡F → (dom ◡F ∈ V → ran ◡F ∈ V))
10		df-rn 2429	. . . . . 6 ⊢ ran F = dom ◡F
11	10	eleq1i 1152	. . . . 5 ⊢ (ran F ∈ V ↔ dom ◡F ∈ V)
12		dfdm4 2525	. . . . . 6 ⊢ dom F = ran ◡F
13	12	eleq1i 1152	. . . . 5 ⊢ (dom F ∈ V ↔ ran ◡F ∈ V)
14	9, 11, 13	3imtr4g 426	. . . 4 ⊢ (Fun ◡F → (ran F ∈ V → dom F ∈ V))
15	7, 14	syl 12	. . 3 ⊢ (∀x ∈ On (F ‘x) ∈ (A ∖ (F “ x)) → (ran F ∈ V → dom F ∈ V))
16	6, 15	mtoi 94	. 2 ⊢ (∀x ∈ On (F ‘x) ∈ (A ∖ (F “ x)) → ¬ ran F ∈ V)
17	2	tz7.48-1 2994	. . 3 ⊢ (∀x ∈ On (F ‘x) ∈ (A ∖ (F “ x)) → ran F ⊆ A)
18		ssexg 1702	. . . 4 ⊢ (A ∈ V → (ran F ⊆ A → ran F ∈ V))
19	18	com12 13	. . 3 ⊢ (ran F ⊆ A → (A ∈ V → ran F ∈ V))
20	17, 19	syl 12	. 2 ⊢ (∀x ∈ On (F ‘x) ∈ (A ∖ (F “ x)) → (A ∈ V → ran F ∈ V))
21	16, 20	mtod 95	1 ⊢ (∀x ∈ On (F ‘x) ∈ (A ∖ (F “ x)) → ¬ A ∈ V)

Syntax hints:  ¬ wn 1   → wi 2   = wceq 1091   ∈ wcel 1092  ∀wral 1201  Vcvv 1348   ∖ cdif 1484   ⊆ wss 1487  Oncon0 2199  ^◡ccnv 2409  dom cdm 2410  ran crn 2411   “ cima 2413  Fun wfun 2416   Fn wfn 2417   ‘cfv 2422

This theorem is referenced by:  tz7.49 2997

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-tp 1814  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

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