Theorem hbf1 2779

Description: Bound-variable hypothesis builder for a one-to-one function.

Hypotheses

Ref	Expression
hbf1.1	⊢ (y ∈ F → ∀x y ∈ F)
hbf1.2	⊢ (y ∈ A → ∀x y ∈ A)
hbf1.3	⊢ (y ∈ B → ∀x y ∈ B)

Assertion

Ref	Expression
hbf1	⊢ (F:A–1-1→B → ∀x F:A–1-1→B)

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

Proof of Theorem hbf1

Step	Hyp	Ref	Expression
1		hbf1.1	. . . 4 ⊢ (y ∈ F → ∀x y ∈ F)
2		hbf1.2	. . . 4 ⊢ (y ∈ A → ∀x y ∈ A)
3		hbf1.3	. . . 4 ⊢ (y ∈ B → ∀x y ∈ B)
4	1, 2, 3	hbf 2751	. . 3 ⊢ (F:A–→B → ∀x F:A–→B)
5	1	hbcnv 2516	. . . 4 ⊢ (y ∈ ◡F → ∀x y ∈ ◡F)
6	5	hbfun 2684	. . 3 ⊢ (Fun ◡F → ∀xFun ◡F)
7	4, 6	hban 704	. 2 ⊢ ((F:A–→B ∧ Fun ◡F) → ∀x(F:A–→B ∧ Fun ◡F))
8		df-f1 2435	. 2 ⊢ (F:A–1-1→B ↔ (F:A–→B ∧ Fun ◡F))
9	8	bial 695	. 2 ⊢ (∀x F:A–1-1→B ↔ ∀x(F:A–→B ∧ Fun ◡F))
10	7, 8, 9	3imtr4 192	1 ⊢ (F:A–1-1→B → ∀x F:A–1-1→B)

Syntax hints:   → wi 2   ∧ wa 196  ∀wal 672   ∈ wcel 1092  ^◡ccnv 2409  Fun wfun 2416  –→wf 2418  –1-1→wf1 2419

This theorem is referenced by:  hbf1o 2798

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-clab 1093  df-cleq 1097  df-clel 1099  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-br 2063  df-opab 2098  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435

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