Theorem hbdm 2565

Description: Bound-variable hypothesis builder for domain.

Hypothesis

Ref	Expression
hbdm.1	⊢ (y ∈ A → ∀x y ∈ A)

Assertion

Ref	Expression
hbdm	⊢ (y ∈ dom A → ∀x y ∈ dom A)

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

Proof of Theorem hbdm

Step	Hyp	Ref	Expression
1		hbdm.1	. . . 4 ⊢ (y ∈ A → ∀x y ∈ A)
2	1	hbcnv 2516	. . 3 ⊢ (y ∈ ◡A → ∀x y ∈ ◡A)
3	2	hbrn 2564	. 2 ⊢ (y ∈ ran ◡A → ∀x y ∈ ran ◡A)
4		dfdm4 2525	. . 3 ⊢ dom A = ran ◡A
5	4	eleq2i 1153	. 2 ⊢ (y ∈ dom A ↔ y ∈ ran ◡A)
6	5	bial 695	. 2 ⊢ (∀x y ∈ dom A ↔ ∀x y ∈ ran ◡A)
7	3, 5, 6	3imtr4 192	1 ⊢ (y ∈ dom A → ∀x y ∈ dom A)

Syntax hints:   → wi 2  ∀wal 672   ∈ wcel 1092  ^◡ccnv 2409  dom cdm 2410  ran crn 2411

This theorem is referenced by:  hbfn 2720

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-cnv 2426  df-dm 2428  df-rn 2429

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