Theorem brelrng 3341

Description: The second argument of a binary relation belongs to its range.

Assertion

Ref	Expression
brelrng	⊢ ((A ∈ F ⋀ B ∈ G ⋀ ACB) → B ∈ ran C)

Proof of Theorem brelrng

Step	Hyp	Ref	Expression
1		breldmg 3314	. . . 4 ⊢ ((B ∈ G ⋀ B◡CA) → B ∈ dom ◡ C)
2	1	3adant1 797	. . 3 ⊢ ((A ∈ F ⋀ B ∈ G ⋀ B◡CA) → B ∈ dom ◡ C)
3		brcnvg 3295	. . . . . 6 ⊢ ((B ∈ G ⋀ A ∈ F) → (B◡CA ↔ ACB))
4	3	ancoms 436	. . . . 5 ⊢ ((A ∈ F ⋀ B ∈ G) → (B◡CA ↔ ACB))
5	4	biimprd 154	. . . 4 ⊢ ((A ∈ F ⋀ B ∈ G) → (ACB → B◡CA))
6	5	3impia 830	. . 3 ⊢ ((A ∈ F ⋀ B ∈ G ⋀ ACB) → B◡CA)
7	2, 6	syld3an3 870	. 2 ⊢ ((A ∈ F ⋀ B ∈ G ⋀ ACB) → B ∈ dom ◡ C)
8		df-rn 3187	. 2 ⊢ ran C = dom ◡ C
9	7, 8	syl6eleqr 1558	1 ⊢ ((A ∈ F ⋀ B ∈ G ⋀ ACB) → B ∈ ran C)

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223   ⋀ w3a 775   ∈ wcel 958   class class class wbr 2617  ^◡ccnv 3167  dom cdm 3168  ran crn 3169

This theorem is referenced by:  brelrn 3342  relelrng 3345  spwpr3OLD 8619

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2701  ax-pow 2740  ax-pr 2777

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-br 2618  df-opab 2665  df-cnv 3184  df-dm 3186  df-rn 3187

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