Theorem cbvfo 2923

Description: Change bound variable between domain and range of function.

Hypothesis

Ref	Expression
cbvfo.1	⊢ ((F ‘x) = y → (φ ↔ ψ))

Assertion

Ref	Expression
cbvfo	⊢ (F:A–onto→B → (∀x ∈ A φ ↔ ∀y ∈ B ψ))

Distinct variable group(s):   x,y,A   x,B,y   x,F,y   φ,y   ψ,x

Proof of Theorem cbvfo

Step	Hyp	Ref	Expression
1		fof 2788	. . 3 ⊢ (F:A–onto→B → F:A–→B)
2		ffun 2754	. . 3 ⊢ (F:A–→B → Fun F)
3		visset 1350	. . . . . . . . . . . 12 ⊢ x ∈ V
4	3	breldm 2535	. . . . . . . . . . 11 ⊢ (xFy → x ∈ dom F)
5	4	a1i 7	. . . . . . . . . 10 ⊢ (Fun F → (xFy → x ∈ dom F))
6		visset 1350	. . . . . . . . . . 11 ⊢ y ∈ V
7	6	funbrfv 2852	. . . . . . . . . 10 ⊢ (Fun F → (xFy → (F ‘x) = y))
8	5, 7	jcad 455	. . . . . . . . 9 ⊢ (Fun F → (xFy → (x ∈ dom F ∧ (F ‘x) = y)))
9	8	19.22dv 947	. . . . . . . 8 ⊢ (Fun F → (∃x xFy → ∃x(x ∈ dom F ∧ (F ‘x) = y)))
10	6	elrn2 2563	. . . . . . . 8 ⊢ (y ∈ ran F ↔ ∃x xFy)
11	9, 10	syl5ib 181	. . . . . . 7 ⊢ (Fun F → (y ∈ ran F → ∃x(x ∈ dom F ∧ (F ‘x) = y)))
12		hba1 698	. . . . . . . 8 ⊢ (∀x(x ∈ dom F → φ) → ∀x∀x(x ∈ dom F → φ))
13		ax-17 925	. . . . . . . 8 ⊢ (ψ → ∀xψ)
14		cbvfo.1	. . . . . . . . . . . 12 ⊢ ((F ‘x) = y → (φ ↔ ψ))
15	14	biimpcd 137	. . . . . . . . . . 11 ⊢ (φ → ((F ‘x) = y → ψ))
16	15	syl3 18	. . . . . . . . . 10 ⊢ ((x ∈ dom F → φ) → (x ∈ dom F → ((F ‘x) = y → ψ)))
17	16	imp3a 279	. . . . . . . . 9 ⊢ ((x ∈ dom F → φ) → ((x ∈ dom F ∧ (F ‘x) = y) → ψ))
18	17	a4s 682	. . . . . . . 8 ⊢ (∀x(x ∈ dom F → φ) → ((x ∈ dom F ∧ (F ‘x) = y) → ψ))
19	12, 13, 18	19.23ad 748	. . . . . . 7 ⊢ (∀x(x ∈ dom F → φ) → (∃x(x ∈ dom F ∧ (F ‘x) = y) → ψ))
20	11, 19	syl9 55	. . . . . 6 ⊢ (Fun F → (∀x(x ∈ dom F → φ) → (y ∈ ran F → ψ)))
21	20	19.21adv 945	. . . . 5 ⊢ (Fun F → (∀x(x ∈ dom F → φ) → ∀y(y ∈ ran F → ψ)))
22	3, 6	brelrn 2559	. . . . . . . . . . 11 ⊢ (xFy → y ∈ ran F)
23	22	a1i 7	. . . . . . . . . 10 ⊢ (Fun F → (xFy → y ∈ ran F))
24	23, 7	jcad 455	. . . . . . . . 9 ⊢ (Fun F → (xFy → (y ∈ ran F ∧ (F ‘x) = y)))
25	24	19.22dv 947	. . . . . . . 8 ⊢ (Fun F → (∃y xFy → ∃y(y ∈ ran F ∧ (F ‘x) = y)))
26	3	eldm 2527	. . . . . . . 8 ⊢ (x ∈ dom F ↔ ∃y xFy)
27	25, 26	syl5ib 181	. . . . . . 7 ⊢ (Fun F → (x ∈ dom F → ∃y(y ∈ ran F ∧ (F ‘x) = y)))
28		hba1 698	. . . . . . . 8 ⊢ (∀y(y ∈ ran F → ψ) → ∀y∀y(y ∈ ran F → ψ))
29		ax-17 925	. . . . . . . 8 ⊢ (φ → ∀yφ)
30	14	biimprcd 138	. . . . . . . . . . 11 ⊢ (ψ → ((F ‘x) = y → φ))
31	30	syl3 18	. . . . . . . . . 10 ⊢ ((y ∈ ran F → ψ) → (y ∈ ran F → ((F ‘x) = y → φ)))
32	31	imp3a 279	. . . . . . . . 9 ⊢ ((y ∈ ran F → ψ) → ((y ∈ ran F ∧ (F ‘x) = y) → φ))
33	32	a4s 682	. . . . . . . 8 ⊢ (∀y(y ∈ ran F → ψ) → ((y ∈ ran F ∧ (F ‘x) = y) → φ))
34	28, 29, 33	19.23ad 748	. . . . . . 7 ⊢ (∀y(y ∈ ran F → ψ) → (∃y(y ∈ ran F ∧ (F ‘x) = y) → φ))
35	27, 34	syl9 55	. . . . . 6 ⊢ (Fun F → (∀y(y ∈ ran F → ψ) → (x ∈ dom F → φ)))
36	35	19.21adv 945	. . . . 5 ⊢ (Fun F → (∀y(y ∈ ran F → ψ) → ∀x(x ∈ dom F → φ)))
37	21, 36	impbid 397	. . . 4 ⊢ (Fun F → (∀x(x ∈ dom F → φ) ↔ ∀y(y ∈ ran F → ψ)))
38		df-ral 1205	. . . 4 ⊢ (∀x ∈ dom Fφ ↔ ∀x(x ∈ dom F → φ))
39		df-ral 1205	. . . 4 ⊢ (∀y ∈ ran Fψ ↔ ∀y(y ∈ ran F → ψ))
40	37, 38, 39	3bitr4g 428	. . 3 ⊢ (Fun F → (∀x ∈ dom Fφ ↔ ∀y ∈ ran Fψ))
41	1, 2, 40	3syl 21	. 2 ⊢ (F:A–onto→B → (∀x ∈ dom Fφ ↔ ∀y ∈ ran Fψ))
42		fdm 2756	. . 3 ⊢ (F:A–→B → dom F = A)
43		raleq 1324	. . 3 ⊢ (dom F = A → (∀x ∈ dom Fφ ↔ ∀x ∈ A φ))
44	1, 42, 43	3syl 21	. 2 ⊢ (F:A–onto→B → (∀x ∈ dom Fφ ↔ ∀x ∈ A φ))
45		forn 2789	. . 3 ⊢ (F:A–onto→B → ran F = B)
46		raleq 1324	. . 3 ⊢ (ran F = B → (∀y ∈ ran Fψ ↔ ∀y ∈ B ψ))
47	45, 46	syl 12	. 2 ⊢ (F:A–onto→B → (∀y ∈ ran Fψ ↔ ∀y ∈ B ψ))
48	41, 44, 47	3bitr3d 423	1 ⊢ (F:A–onto→B → (∀x ∈ A φ ↔ ∀y ∈ B ψ))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = wceq 1091   ∈ wcel 1092  ∀wral 1201   class class class wbr 2054  dom cdm 2410  ran crn 2411  Fun wfun 2416  –→wf 2418  –onto→wfo 2420   ‘cfv 2422

This theorem is referenced by:  cbvexfo 2924  isowe 2941  f1oweOLD 2944

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-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  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-br 2063  df-opab 2098  df-id 2125  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-fo 2436  df-fv 2438

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