Theorem hbf 2751

Description: Bound-variable hypothesis builder for a mapping.

Hypotheses

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

Assertion

Ref	Expression
hbf	⊢ (F:A–→B → ∀x F:A–→B)

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

Proof of Theorem hbf

Step	Hyp	Ref	Expression
1		hbf.1	. . . 4 ⊢ (y ∈ F → ∀x y ∈ F)
2		hbf.2	. . . 4 ⊢ (y ∈ A → ∀x y ∈ A)
3	1, 2	hbfn 2720	. . 3 ⊢ (F Fn A → ∀x F Fn A)
4	1	hbrn 2564	. . . 4 ⊢ (y ∈ ran F → ∀x y ∈ ran F)
5		hbf.3	. . . 4 ⊢ (y ∈ B → ∀x y ∈ B)
6	4, 5	hbss 1501	. . 3 ⊢ (ran F ⊆ B → ∀xran F ⊆ B)
7	3, 6	hban 704	. 2 ⊢ ((F Fn A ∧ ran F ⊆ B) → ∀x(F Fn A ∧ ran F ⊆ B))
8		df-f 2434	. 2 ⊢ (F:A–→B ↔ (F Fn A ∧ ran F ⊆ B))
9	8	bial 695	. 2 ⊢ (∀x F:A–→B ↔ ∀x(F Fn A ∧ ran F ⊆ B))
10	7, 8, 9	3imtr4 192	1 ⊢ (F:A–→B → ∀x F:A–→B)

Syntax hints:   → wi 2   ∧ wa 196  ∀wal 672   ∈ wcel 1092   ⊆ wss 1487  ran crn 2411   Fn wfn 2417  –→wf 2418

This theorem is referenced by:  hbf1 2779  fopab2 2891

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

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