Theorem tz7.49c 2998

Description: Corollary of Proposition 7.49 of [TakeutiZaring] p. 51.

Hypotheses

Ref	Expression
tz7.48.1	⊢ F Fn On
tz7.49.2	⊢ A ∈ V

Assertion

Ref	Expression
tz7.49c	⊢ (∀x ∈ On (¬ (A ∖ (F “ x)) = ∅ → (F ‘x) ∈ (A ∖ (F “ x))) → ∃x ∈ On (F ↾ x):x–1-1-onto→A)

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

Proof of Theorem tz7.49c

Step	Hyp	Ref	Expression
1		tz7.48.1	. . 3 ⊢ F Fn On
2		tz7.49.2	. . 3 ⊢ A ∈ V
3	1, 2	tz7.49 2997	. 2 ⊢ (∀x ∈ On (¬ (A ∖ (F “ x)) = ∅ → (F ‘x) ∈ (A ∖ (F “ x))) → ∃x ∈ On (∀y ∈ x ¬ (A ∖ (F “ y)) = ∅ ∧ (F “ x) = A ∧ Fun ◡(F ↾ x)))
4		onsst 2243	. . . . . . . . . 10 ⊢ (x ∈ On → x ⊆ On)
5		fnssres 2734	. . . . . . . . . . 11 ⊢ ((F Fn On ∧ x ⊆ On) → (F ↾ x) Fn x)
6	1, 5	mpan 518	. . . . . . . . . 10 ⊢ (x ⊆ On → (F ↾ x) Fn x)
7	4, 6	syl 12	. . . . . . . . 9 ⊢ (x ∈ On → (F ↾ x) Fn x)
8		df-ima 2431	. . . . . . . . . . 11 ⊢ (F “ x) = ran (F ↾ x)
9	8	cleq1i 1108	. . . . . . . . . 10 ⊢ ((F “ x) = A ↔ ran (F ↾ x) = A)
10	9	biimp 133	. . . . . . . . 9 ⊢ ((F “ x) = A → ran (F ↾ x) = A)
11	7, 10	anim12i 268	. . . . . . . 8 ⊢ ((x ∈ On ∧ (F “ x) = A) → ((F ↾ x) Fn x ∧ ran (F ↾ x) = A))
12	11	anim1i 269	. . . . . . 7 ⊢ (((x ∈ On ∧ (F “ x) = A) ∧ Fun ◡(F ↾ x)) → (((F ↾ x) Fn x ∧ ran (F ↾ x) = A) ∧ Fun ◡(F ↾ x)))
13		f1o2 2804	. . . . . . . 8 ⊢ ((F ↾ x):x–1-1-onto→A ↔ ((F ↾ x) Fn x ∧ Fun ◡(F ↾ x) ∧ ran (F ↾ x) = A))
14		df-3an 583	. . . . . . . 8 ⊢ (((F ↾ x) Fn x ∧ Fun ◡(F ↾ x) ∧ ran (F ↾ x) = A) ↔ (((F ↾ x) Fn x ∧ Fun ◡(F ↾ x)) ∧ ran (F ↾ x) = A))
15		an23 371	. . . . . . . 8 ⊢ ((((F ↾ x) Fn x ∧ Fun ◡(F ↾ x)) ∧ ran (F ↾ x) = A) ↔ (((F ↾ x) Fn x ∧ ran (F ↾ x) = A) ∧ Fun ◡(F ↾ x)))
16	13, 14, 15	3bitr 155	. . . . . . 7 ⊢ ((F ↾ x):x–1-1-onto→A ↔ (((F ↾ x) Fn x ∧ ran (F ↾ x) = A) ∧ Fun ◡(F ↾ x)))
17	12, 16	sylibr 175	. . . . . 6 ⊢ (((x ∈ On ∧ (F “ x) = A) ∧ Fun ◡(F ↾ x)) → (F ↾ x):x–1-1-onto→A)
18	17	exp31 293	. . . . 5 ⊢ (x ∈ On → ((F “ x) = A → (Fun ◡(F ↾ x) → (F ↾ x):x–1-1-onto→A)))
19	18	imp3a 279	. . . 4 ⊢ (x ∈ On → (((F “ x) = A ∧ Fun ◡(F ↾ x)) → (F ↾ x):x–1-1-onto→A))
20		3simpc 593	. . . 4 ⊢ ((∀y ∈ x ¬ (A ∖ (F “ y)) = ∅ ∧ (F “ x) = A ∧ Fun ◡(F ↾ x)) → ((F “ x) = A ∧ Fun ◡(F ↾ x)))
21	19, 20	syl5 22	. . 3 ⊢ (x ∈ On → ((∀y ∈ x ¬ (A ∖ (F “ y)) = ∅ ∧ (F “ x) = A ∧ Fun ◡(F ↾ x)) → (F ↾ x):x–1-1-onto→A))
22	21	r19.22i 1273	. 2 ⊢ (∃x ∈ On (∀y ∈ x ¬ (A ∖ (F “ y)) = ∅ ∧ (F “ x) = A ∧ Fun ◡(F ↾ x)) → ∃x ∈ On (F ↾ x):x–1-1-onto→A)
23	3, 22	syl 12	1 ⊢ (∀x ∈ On (¬ (A ∖ (F “ x)) = ∅ → (F ‘x) ∈ (A ∖ (F “ x))) → ∃x ∈ On (F ↾ x):x–1-1-onto→A)

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196   ∧ w3a 581   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202  Vcvv 1348   ∖ cdif 1484   ⊆ wss 1487  ∅c0 1707  Oncon0 2199  ^◡ccnv 2409  ran crn 2411   ↾ cres 2412   “ cima 2413  Fun wfun 2416   Fn wfn 2417  –1-1-onto→wf1o 2421   ‘cfv 2422

This theorem is referenced by:  numthlem 3598

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-int 1966  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-fo 2436  df-f1o 2437  df-fv 2438

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